pricing american options by simulation by cem...

PRICING AMERICAN OPTIONS BY SIMULATION

by

Cem Coskan

B.S., Management Engineering, Istanbul Technical University, 2006

Submitted to the Institute for Graduate Studies in

Science and Engineering in partial fulfillment of

the requirements for the degree of

Master of Science

Graduate Program in Industrial Engineering

Bogazici University

2008

ii


APPROVED BY:

Assoc. Prof. Wolfgang Hormann . . . . . . . . . . . . . . . . . . .

(Thesis Supervisor)

Prof. Mehmet Bolak . . . . . . . . . . . . . . . . . . .

Prof. Refik Gullu . . . . . . . . . . . . . . . . . . .

DATE OF APPROVAL: 10.06.2008

iii

ACKNOWLEDGEMENTS

It is difficult to state my gratitude to my supervisor, Assoc. Prof. Wolfgang

Hormann. Without his inspiration and great efforts to explain things clearly and

simply, I would probably be lost in thesis study. Besides introducing me to the field of

computational finance, I am also grateful for his advices and guidance throughout my

thesis period.

I am deeply indebted to Assist. Prof. Mehtap Hisarcıklılar from Istanbul Tech-

nical University who helped and encouraged me not only while writing this thesis, but

also since I have began the graduate program. In addition, I am thankful to TUBITAK

for the financial assistance that is provided until my graduation.

During this work, I have collaborated with many colleagues for whom I have great

regard and I want to express that I thank much to those who have helped me with my

work besides providing a stimulating and fun environment.

Finally, I wish to extend my warmest thanks to my parents for their suppoert

and to all who really care about me.

iv

ABSTRACT


Pricing American options is a difficult problem in mathematical finance as no

closed form solution exists. So, many approximations and numerical techniques have

been developed. In option pricing, it is advantageous to use Monte Carlo simulation

due to its precision and convergence. In this thesis, we implement and analyze the

Least Squares Monte Carlo (LSM) algorithm of Longstaff and Schwartz (2001), which

is a regression-based method for American option pricing. For one-dimensional cases,

we offer an alternative method that requires less computational effort while the ac-

curacy is increased; Simple Regression Approach (SRA), which is based on the LSM

algorithm. Moreover, we analyze the options on multi-assets and use the LSM algo-

rithm to price these options. As well as applying variance reduction techniques and

reaching a considerable decrease in the estimated variance in multi-dimensional cases,

we also investigate how many early exercise opportunities should be considered in the

algorithm for the option to be close to American one. Numerical results indicate that

the price estimates in the literature are actually low for American options on the max-

imum of two and five assets. In addition, we determine a useful set of basis functions

for pricing American spread options. For all implementations we use R, which is a

programming language and environment for statistical computing and graphics.

v

OZET

AMERIKAN OPSIYONLARININ SIMULASYONLA

FIYATLANDIRILMASI

Matematiksel finansta, Amerikan opsiyonlarının fiyatlandırılması kapalı formul

cozumu bulunmadıgı icin zor bir problemdir. Bu sebeple, bircok yaklasım yontemleri

ve sayısal teknikler gelistirilmistir. Opsiyon fiyatlandırmada Monte Carlo simulasyonu

kullanmak, hassasiyetinden ve yakınsamasından oturu avantajlıdır. Bu tezde, Amerikan

opsiyonlarının fiyatlandırılması icin Longstaff ve Schwartz’ın (2001), regresyona dayalı

bir metot olan, En Kucuk Kareler Monte Carlo algoritmasını uyguladık ve analiz ettik.

Tek boyutlu ornekler icin dogruluk artarken daha az hesaplama gucune ihtiyac duyan,

En Kucuk Kareler Monte Carlo algoritmasına dayanan, Basit Regresyon Yaklasımı adlı

alternatif bir yontem onerdik. Ayrıca, birden fazla varlıga dayalı opsiyonları analiz ettik

ve bu opsiyonları fiyatlandırmak icin En Kucuk Kareler Monte Carlo algoritmasını kul-

landık. Cok boyutlu orneklerde varyans azaltma teknikleri uygulamanın ve varyansta

kayda deger bir dusus saglamanın yanı sıra, opsiyonun Amerikan opsiyonuna yakın ola-

bilmesi icin kac adet erken kullanım fırsatının goz onunde bulundurulması gerektigini

arastırdık. Sayısal sonuclar iki ve bes varlıgın maksimum fiyatına baglı opsiyonların

literaturdeki fiyat tahminlerinin aslında dusuk olduguna isaret etmektedir. Ek olarak,

Amerikan yayılma opsiyonları icin yararlı bir temel fonksiyon kumesi belirledik. Tum

uygulamalar icin programlama dili ve istatiksel hesaplama-cizgeleme ortamı olan R’ı

kullandık.

vi

TABLE OF CONTENTS

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

OZET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

LIST OF SYMBOLS/ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . xii

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. OPTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2. Types of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3. American Options and Early Exercise . . . . . . . . . . . . . . . . . . . 5

3. FUNDAMENTALS IN OPTION PRICING . . . . . . . . . . . . . . . . . . 7

3.1. Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2. Risk Neutral Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3. Option Pricing by Monte Carlo Simulation . . . . . . . . . . . . . . . . 10

3.3.1. Variance Reduction Techniques . . . . . . . . . . . . . . . . . . 10

4. PRICING AMERICAN OPTIONS . . . . . . . . . . . . . . . . . . . . . . . 14

4.1. Methods in the Literature . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2. Simulation of American Options . . . . . . . . . . . . . . . . . . . . . . 16

4.3. LSM algorithm of Longstaff and Schwartz . . . . . . . . . . . . . . . . 17

5. ASSESSMENT OF LSM ALGORITHM . . . . . . . . . . . . . . . . . . . . 21

5.1. Implementation of LSM in R . . . . . . . . . . . . . . . . . . . . . . . . 21

5.2. How many basis functions are necessary? . . . . . . . . . . . . . . . . . 23

5.3. Problems in LSM algorithm . . . . . . . . . . . . . . . . . . . . . . . . 24

5.3.1. Inefficiency of Regression . . . . . . . . . . . . . . . . . . . . . . 24

5.3.2. Boundary Estimation . . . . . . . . . . . . . . . . . . . . . . . . 26

5.4. Improvement of Region of Regression . . . . . . . . . . . . . . . . . . . 28

5.4.1. The Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.4.2. R Implementation of SRA . . . . . . . . . . . . . . . . . . . . . 30

vii

5.4.3. Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 31

6. OPTIMIZATION APPROACH . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.1. R Implementation of Optimization Approach . . . . . . . . . . . . . . . 33

6.2. Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7. MULTI-ASSET OPTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.1. Types of Multi-Asset Options . . . . . . . . . . . . . . . . . . . . . . . 37

7.1.1. Options on the Maximum of k Assets . . . . . . . . . . . . . . . 37

7.1.2. Spread Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.2. Simulation of multi-asset options . . . . . . . . . . . . . . . . . . . . . 42

7.2.1. Cholesky decomposition . . . . . . . . . . . . . . . . . . . . . . 42

8. PRICING MULTI-ASSET AMERICAN OPTIONS . . . . . . . . . . . . . . 44

8.1. Simulation of American-style Multi-asset Options . . . . . . . . . . . . 44

8.2. LSM Algorithm for Multi-Asset American Options . . . . . . . . . . . 45

8.3. American Options on the Maximum of Two Assets . . . . . . . . . . . 46

8.3.1. Price of European Option on the Maximum of Two Assets as

Control Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

8.3.2. R Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 46


8.4. American Options on the Maximum of Five Assets . . . . . . . . . . . 53


8.5. American Spread Options . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.5.1. European Spread Option Price as Control Variable . . . . . . . 57

8.5.2. R Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 58


9. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

APPENDIX A: R CODES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

A.1. R Codes for Black-Scholes Formula to Price European Options . . . . . 67

A.2. R Codes for the Implementation of the LSM Algorithm . . . . . . . . . 67

A.3. R Codes for the Exact Early Exercise Boundary of an American Put at

time td−1 by Newton-Raphson Method . . . . . . . . . . . . . . . . . . . 69

A.4. R Codes for the Implementation of the SRA . . . . . . . . . . . . . . . 69

A.5. R Codes for the Implementation of the Optimization Approach . . . . 71

viii

A.6. R Codes for the Exact Price of a European Call on the Maximum of

Two Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A.7. R Codes of the Kirk’s approximation for European Spread Options . . 73

A.8. R Codes to Price an American Call on the Maximum of Two Assets

without Pilot Study for CV . . . . . . . . . . . . . . . . . . . . . . . . . 73


with Pilot Study for CV . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

A.10.R Codes to Price an American Call on the Maximum of Five Assets . . 79

A.11.R Codes to Price a European Spread Call . . . . . . . . . . . . . . . . 82

A.12.R Codes to Price an American Spread Call without Pilot Study for CV 83

A.13.R Codes to Price an American Spread Call with Pilot Study for CV . . 84

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

ix

LIST OF FIGURES

Figure 5.1. Plot of LSM estimates of continuation values versus immediate cash

flows from early exercise at time td−1 . . . . . . . . . . . . . . . . 25

Figure 6.1. Plot of total values for different early exercise thresholds at time td−1 36

Figure 8.1. Early exercise region of an American call option on the maximum

of two assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Figure 8.2. Total number of time steps versus early exercise premium for an

American call option on the maximum of two assets . . . . . . . . 52


American call option on the maximum of five assets . . . . . . . . 56

Figure 8.4. Early exercise region of an American spread call option at time td−1 63

Figure 8.5. Plots of early exercise premium versus number of time steps for

American spread call options . . . . . . . . . . . . . . . . . . . . . 64

x

LIST OF TABLES

Table 5.1. Price estimates for different number of basis functions and varied

initial stock price, volatility and lifetime . . . . . . . . . . . . . . . 24

Table 5.2. Early exercise boundary calculated by LSM algorithm for a Bermu-

dan option with two exercise opportunities and comparison with

exact values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Table 5.3. Early exercise boundary calculated by SRA for a Bermudan option

with two exercise opportunities and comparison with exact values . 32

Table 5.4. Comparison of prices calculated by SRA, LSM and FD, and TRFs

varying initial stock price, volatility and lifetime . . . . . . . . . . 32

Table 6.1. Prices calculated by optimization approach, LSM and FD, varying

initial stock price, volatility and lifetime . . . . . . . . . . . . . . . 35

Table 8.1. Price estimates for a two asset American maximum call option . . 49

Table 8.2. Price estimates of binomial tree and LSM with different number of

time steps for a two asset American maximum call option on two

assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Table 8.3. Price estimates for an American call option on the maximum of five

assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Table 8.4. Price estimates for an American call option on the maximum of five

assets with different number of time steps . . . . . . . . . . . . . . 55

xi

Table 8.5. Price estimates of European spread call using Binomial tree, Kirk’s

approximation and Monte Carlo simulation . . . . . . . . . . . . . 58

Table 8.6. Sets of basis functions used in pricing an American spread call . . 60

Table 8.7. Price estimates for American spread call option by using different

sets of basis functions in pricing algorithm . . . . . . . . . . . . . . 61

Table 8.8. VRFs in pricing American spread options . . . . . . . . . . . . . . 62

xii

LIST OF SYMBOLS/ABBREVIATIONS

b Cost of carry

d Number of time steps

h Payoff function

k Number of underlying assets

K Strike price

n Number of generated paths

Qi Quantity of the ith asset

r Risk-free interest rate

S0 Initial asset price

Si0 Initial price of the ith asset

St Price of the asset at time t

Sit Price of the ith asset at time t

t Time

T Maturity time

Vt Value of an option at time t

β Regression coefficients

δ Continous dividend yield

ψ Basis functions

ρ Correlation

σ Volatility

τ Stopping time

AV Antithetic variates

BS Black-Scholes

CI Confidence interval

CV Control variates

exp Exponential power

FD Finite difference

xiii

GBM Geometric Brownian motion

i.i.d independent and identically distributed

LSM Least squares Monte Carlo

max Maximum of

NR Newton-Raphson

OPT Optimization

PDE Partial differential equation

SDE Stochastic differential equation

SRA Simple regression approach

TRF Time reduction factor

VRF Variance reduction factor

1

1. INTRODUCTION

An option is a financial instrument which gives its holder the right to receive a

contingent payoff. There are several types of options. One common type is American

options, which allow the holder to receive payoff at any time during the lifetime of the

option. Hence, as expected, it is not easy to price this type of option. Although some

derivatives, such as European options, can be priced analytically, a general closed-form

analytical solution for the evaluation of American options does not exist.

The use of Monte Carlo simulation in option pricing is advantageous regarding its

precision and convergence, which is generally independent of the number of underlying

variables. However, it has the disadvantage of being quite slow. In the last decade,

several articles have been published on how to price American options by Monte Carlo

simulation (Andersen and Broadie, 2004; Broadie and Glasserman, 2004; Longstaff and

Schwartz, 2001; Tsitsiklis and Van Roy, 2001). Monte Carlo methods usually simulate

the development of the underlying asset with random numbers, to determine several

possible price movements, subject to the known market data.

The aim of this thesis study is to understand and implement the pricing of Amer-

ican options by simulation. We focus on the Least Squares Monte Carlo (LSM) method

of Longstaff and Schwartz (2001) to price American options, analyze and try to im-

prove it in terms of computational cost while increasing its accuracy. Also, we apply

the LSM algorithm together with variance reduction techniques in pricing multidimen-

sional American options and investigate how many time points should be considered

within the simulation. We make all the implementations in R, which is a free software

environment for statistical computing and graphics.

The organisation of the thesis is as follows: First, Chapter 2 gives general infor-

mation about options. Then, fundamentals in option pricing are introduced to reveal

the assumptions and to comprehend the use of simulation in option pricing in Chapter

3. In Chapter 4, the literature on different approaches to price American options is

2

summarized. In addition, it describes the issues about the simulation of American

options and the Least Squares Monte Carlo (LSM) method of Longstaff and Schwartz

(2001). In Chapter 5, we assess the LSM algorithm with a number of numerical exper-

iments and introduce an alternative method to price American options, named simple

regression approach (SRA), in order to increase the efficiency of the LSM algorithm.

An optimization approach is implemented and its results are discussed in Chapter 6.

Then, in Chapter 7, multi-asset American options are explained. Finally, we price

American options on the maximum of two or five assets and American spread options,

both of which are classified as multi-asset American options, by applying variance re-

duction techniques and give numerical results in Chapter 8. Conclusions, Appendix

and references are added at the end of the thesis.

3

2. OPTIONS

In this chapter, options are described in general. After that some different types

of options are explained. Besides, important issues about early exercise of American

options are stated. Hull (2003) and Lyuu (2002) are references commonly used in this

chapter.

2.1. Definition

In the world of finance, a financial security whose value is derived from another

underlying financial security is called as a derivative (Hull, 2003). Derivatives can be

used to hedge risk as well as for speculative purposes.

An option, which is also a financial derivative, represents a contract between

two parties, which is sold by one party to another and gives the right, but not the

obligation, to buy or to sell a security or an asset at an agreed-upon exercise (strike)

price (Hull, 2003).

According to the Options Industry Council (OIC), which is a Chicago-based non-

profit industry group, the options marketplace was slack and unregulated until the

mid 20th century (OIC, 2008). The Chicago Board Options Exchange (CBOE) began

trading options in 1973 (Lyuu, 2002). As stated by the 2006 Market Statistics report of

CBOE, more than two billion option contracts changed hands in 2006. In this volume of

the market, CBOE had a share of 33 per cent, which is worth almost 311 billion dollars.

Furthermore, Clary (2007) argues that the volume of the options has already exceeded

the record occurred in 2006. It is also asserted that Wall Street tries to convince

people to use options. On the other hand, the marketplace of options in Turkey, which

is named as Turkish Derivative Exchange (TurkDEX), started operations in 2001 and

the volume of the operations is increasing rapidly year by year (TurkDEX, 2008).

4

2.2. Types of Options

The general distinction in options is on the right that is gained. If the option

owner gains the right to purchase, this option is called a call option. Oppositely, the

type of option that gives the right to sell is named put option. The payoff function h

is:

hc(S) = max(S −K, 0) = (S −K)+ for calls

hp(S) = max(K − S, 0) = (K − S)+ for puts

where S is the price of the underlying security and K is the exercise (strike) price.

Zero indicates that the option is not exercised.

According to when options can be exercised, they are classified into three groups;

European, Bermudan and American Options. It is possible to exercise American op-

tions at any time within the time horizon whereas European options only are exercised

at the pre-defined expiry date. Therefore, when the price of the underlying is high

enough for calls and low enough for puts, which results payoff to be high, it can influ-

ence its owner to exercise the American option early (Charnes, 2000). Early exercise

refers to the exercise of the option before maturity. Bermudan options are similar in

style to American options regarding the possibility of early exercise. However, the dif-

ference is the fact that a Bermudan option has predetermined discrete exercise dates.

So, Bermudan options can be placed between European and American options (Glasser-

man, 2003).

Additionally, Exotic options differ from common options in terms of the under-

lying or the calculation of the payoff. Asian, Barrier, Rainbow, Spread, Basket and

maximum options are some commonly traded options that are categorized as exotic

ones.

5

2.3. American Options and Early Exercise

The basic feature of American options is that it enables the owner to exercise

earlier than maturity date (Hull, 2003). In this section, the early exercise of American

calls and puts is explained.

Lyuu (2002) asserts that an American call option on a non-dividend paying stock

should not be exercised early. This can be shown by an example. Let us consider two

portfolios. Portfolio A is composed of a bond that has an initial value of Ke−rT and

an American call on a non-dividend paying stock whereas Portfolio B is only one share

of the underlying stock. T is the expiration date and r is the risk-free interest rate. If

early exercise occurs at a time t, such that 0 < t < T , the value of Portfolio A would

be:

Ke−r(T−t) + St −K = St −K(1− e−r(T−t)

)≤ St

and of Portfolio B would be St which denotes the stock price at time t. It can be

noticed that if the call is exercised early, worth of Portfolio A is smaller than or equal

to the worth of Portfolio B. However, on the maturity date T , Portfolio B would have

a value of ST while Portfolio A would have:

K + max(ST −K, 0) = max(K,ST )

Therefore, if the option were held until expiration, Portfolio A would be worth at

least as much as Portfolio B. This shows that when the underlying stock is paying no

dividend, an American call should not be exercised early. In addition, it implies that

an American call has the same value as a European call option. This does not mean

that we must keep an American call until the expiration date. Selling it may be a

better alternative.

Exercise of an American call option on a dividend paying stock should take place

at maturity date or before the stock goes ex-dividend (Hull, 2003). As the right to

6

collect dividend will be obtained, it may be optimal to exercise it before the ex-dividend

date. When we consider the same two portfolios mentioned above, it can happen that

by the effect of dividend payment at early exercise time t, portfolio A may have a

greater value than portfolio B.

On the other hand, early exercise of an American put might be optimal regardless

of whether the underlying stock is paying dividend or not (Lyuu, 2002). Hull (2003)

explains this fact by considering two portfolios. Let Portfolio C consist of an American

put option plus one share of the underlying stock and Portfolio D consist of a bond

that has an initial value of K exp(−rT ). At time t, where 0 < t < T , the worth of

Portfolio D amounts to K exp(−r(T−t)) if the option is exercised, Portfolio C is worth

K, which will be the worth of Portfolio D at maturity date. If it is not exercised at

time t, the worth of Portfolio C is max(K,ST ) at maturiy. Lyuu emphasizes that early

exercise is preferred due to the time value of money because early exercise results in

an instant income of K, which can be reinvested.

In short, an American call on a non-dividend paying stock can be evaluated

as European option. Therefore, in the one-dimensional case, we consider and price

American put options instead of call options. Furthermore, due to the feature of early

exercise, it is clear that an American option must have a price at least as high as its

European equivalent.

7

3. FUNDAMENTALS IN OPTION PRICING

This chapter covers the model of Black and Scholes (1973), risk neutral valuation

and the use of simulation which are assumed to be elements of fundamentals in option

pricing.

3.1. Black-Scholes Model

Black and Scholes (1973) built the derivative pricing theory based on geometric

Brownian motion (GBM). The price of an asset at time t, St, is said to follow a GBM

if it satisfies the stochastic differential equation (SDE):

dSt = µStdt+ σStdWt (3.1)

where Wt is the Wiener process or Brownian motion, µ the percentage drift and σ the

percentage volatility, are constants. A Brownian motion Wt is characterized by the

following facts:

• W0 = 0

• Wt is almost surely continuous

• for 0 ≤ s < t, the independent increments Wt−Ws are normally distributed with

mean zero and variance (t− s).

Notice that Equation 3.1 can be written in the following way (the left hand side is the

return of the asset):

dStSt

= µdt+ σdWt

By the help of Ito’s lemma, the analytic solution of Equation 3.1 is

St = S0 exp(

(µ− σ2/2)t+ σ√tZ)

(3.2)

8

for an arbitrary initial value S0, where Z is a standard normal variate (Lyuu, 2002).

Hence, the log-returns for St are normally distributed with mean (µ − σ2/2)t and

variance, σ2t.

Under the assumptions of Black-Scholes, the Black-Scholes PDE, which holds

whenever V , the price of the derivative, is twice differentiable with respect to S, is

∂V

∂t+

1

2σ2S2∂

2V

∂S2+ rS

∂V

∂S− rV = 0 (3.3)

by arbitrage-free pricing. Further details can be seen in Chapter 15 of Lyuu (2002).

The Black-Scholes formula (Black and Scholes, 1973), which is driven by equation

3.3, gives the exact value of a European call or put option, whereas American options

do not have any closed form solution. The price of European call V EUc and European

put V EUp on a non-dividend paying asset, currently trading at price S can be calculated

by the Black-Scholes formula:

V EUc = SN (d1)−Ke−rTN (d2) (3.4)

V EUp = Ke−rTN (−d2)− SN (−d1) (3.5)

where N(.) denotes the cumulative distribution function of the standard normal dis-

tribution, T is the lifetime of the option and

d1 =ln (S/K) + (r + σ2/2)T

σ√T

(3.6)

d2 = d1 − σ√T (3.7)

Implementation of the Black-Scholes formula in R can be seen in Appendix A.1.

9

3.2. Risk Neutral Valuation

Cox and Ross (1976) suggest risk neutrality in derivative pricing. According to

risk neutrality valuation derivatives can be valued under the assumption that investors

are risk neutral (Hull, 2003). This means that the risk preferences of the investors have

no effect on the value. In a risk-neutral world, there are two simple results:

i. The expected return from all securities is the risk-free interest rate.

ii. The appropriate discount rate for any expected future cash flow is the risk-free

interest rate.

Hence, the price of the underlying follows the risk-neutral process,

dSt = rStdt+ σStdWt

where the drift µ in Equation 3.1 is replaced by the constant risk free interest rate r if

the underlying is paying no dividend (Lyuu, 2002). Then, the price of this underlying

asset at time t, St, can be derived by

St = S0 exp(

(r − σ2/2)t+ σ√tZ)

(3.8)

Furthermore, V EU , which is the price of a European option that expires at time T is

given by the discounted expected payoff of the option with the risk-free interest rate,

V EU = E[e−rTh(ST )

]where ST is the value of the underlying asset at the option expiry date. In addition,

the value of the American option can be written as:

V US = supτ

E[e−rτh(Sτ )]

where the maximum is taken over all stopping times 0 ≤ τ ≤ T .

10

3.3. Option Pricing by Monte Carlo Simulation

Monte Carlo methods are a class of computational algorithms that are based on

repeated computation and random sampling (Glasserman, 2003). Monte Carlo simu-

lation is used in finance to value and analyze instruments, portfolios and investments

by simulating the sources of uncertainity that affects their value. Then the value is

determined according to the simulated underlying inputs.

Options can be priced by Monte Carlo simulation (Glasserman, 2003). First, the

price of the underlying asset is simulated by random number generation for a number

of paths as in Equation 3.2. Then, the value of the option is found by calculating

the average of discounted returns over all paths. Since the option is priced under risk

neutral measure, the discount rate is the risk-free interest rate.

In order to get a good estimate from simulation (Glasserman, 2003), the variance

of the estimator should go to zero and thus the number of samples goes to infinity,

which is impossible. Therefore, variance reduction techniques help us to obtain a

better estimate in simulation. In the next section, variance reduction techniques that

are used throughout the thesis are summarized.

3.3.1. Variance Reduction Techniques

Simulations driven by random inputs will produce random outputs (Glasserman,

2003). If we can somehow reduce the variance of an output random variable of interest

without disturbing its expectation, we should have greater precision that means shorter

confidence intervals. In Monte Carlo simulation, it is possible to reduce the variance by

using a number of variance reduction techniques. The brief explanation of two variance

reduction techniques, namely Antithetic Variates (AV) and Control Variates (CV) can

be found in this section.

11

3.3.1.1. Antithetic Variates (AV). The method of antithetic variates (AV) is a vari-

ance reduction technique, which introduces negative dependence between pairs of repli-

cations (Glasserman, 2003). For instance, pairing a sequence Z1, Z2, . . . , Zn of i.i.d.

N(0, 1) variables with the sequence −Z1,−Z2, . . . ,−Zn of i.i.d N(0, 1) variables is the

implementation of AV for a simulation driven by independent standard normal random

variables.

Let us suppose we would like to estimate µ = E [A], where A is the output of

a simulation experiment, and we have generated two samples, A1 and A2. Then an

unbiased estimate of µ is given by:

µ =1

n

(A1 + A2

2

)

as n, which is the number of paths that each sample has, goes to infinity. Then the

variance of the AV estimator is:

V ar (µ) =V ar (A1) + V ar (A2) + 2Cov (A1, A2)

4

Since the variances of the samples equal to each other, the variance of the estimator

becomes:

V ar (µ) =V ar (A1) + Cov (A1, A2)

2

The variance will be reduced since we generate pairs such that

Cov (A1, A2) < 0

Further details can be found in Chapter 4 of Glasserman (2003).

3.3.1.2. Control Variates (CV). Suppose again that we wish to estimate µ = E [A]

where A is the output of a simulation experiment. Thus, our usual estimator is µ = A.

12

Assume that another output of the simulation is X and we know E [X]. Then for any

fixed c,

µc = A− c (X − E (X)) (3.9)

is a control variate (CV) estimator where the observed error (X − E (X)) serves as a

control on the estimate (Glasserman, 2003). In other words, X is a control variable for

A. By Equation 3.9, the variance of the CV estimator becomes:

V ar (µc) = V ar (A) + c2V ar(X)− 2cCov(X,A) (3.10)

Notice that if c equals to zero, then

V ar (µc) = V ar (µ)

Hence, the CV estimator has smaller variance than the standard estimator if

c2V ar (X) < 2cCov(X,A)

The optimal coefficient c∗, which minimizes the variance of the CV estimator, can be

calculated by

c∗ =Cov(X,A)

V ar(X)(3.11)

Substituting this value into Equation 3.10, we find that

V ar (µc) = V ar (µ)− Cov(X,A)2

V ar(X)

As long as we find an X such that Cov(X,A) is not equal to zero, a variance reduction

will be achieved.

Generally Cov(X,A) and V ar(X) are not known. It is possible to estimate c* by

13

using the information within the simulation. However, this results in a biased estimate.

A small pilot study is enough to remove that bias.

14

4. PRICING AMERICAN OPTIONS

This section gives a summary of methods in the literature for pricing American

options and provides clarification of several important issues in simulation of American

options. Also, the simulation method of Longstaff and Schwartz is explained briefly.

4.1. Methods in the Literature

Basic pricing methods for American options are approximation techniques, finite

difference (FD) methods, binomial trees and Monte Carlo simulation.

One of the earliest approximation methods is the work of Geske (1979). He

approximated the price of the American option by considering a compound option

with two exercise dates. In 1984, besides improving the accuracy by considering more

exercisable dates, Geske and Johnson introduced the use of Richard extrapolation in

option pricing. The quadratic approximation of Barone-Adesi and Whaley (1987),

which gives accurate results only for short periods, is another early approach. Several

approximation techniques are compared in Broadie and Detemple (1996) and AitSahlia

and Carr (1997).

FD method is a traditional method that directly deals with the pricing equa-

tions. When the continuous time and space variables are discretized with constant

step lengths, a FD grid is formed. Then, the derivative is priced at each node starting

with the terminal nodes and proceeding recursively up until the starting time through

the FD scheme, which is obtained by discrete approximations to the PDE 3.3. Tavella

(2002) emphasizes that the ability to capture early exercise effectively is an advantage

of the FD method. An early example to the application of the FD method on pricing

derivatives is the work of Brennan and Schwartz (1977).

Binomial tree, which was first proposed by Cox et al. (1979), is another method

to value options. American options can be valued after approximating the underlying

15

stochastic process as a lattice by this technique. In 1996, Broadie and Detemple sug-

gested improving the binomial method by using the analytic Black Scholes formula and

reached a faster rate of convergence. Broadie and Yamamoto (2005) applied the Fast

Gauss Transform to speed up calculations to price American options by the binomial

method.

Simulation acts as a promising alternative to the methods mentioned previously.

An important simulation method for American options was Tilley (1993). This en-

couraged people to investigate the possibility of pricing American options using Monte

Carlo methods. Boyle et al. (1997) reviewed this and other early methods such as

Barraquand and Martineau (1995), Bossaerts (1989), Broadie and Glasserman (1997).

To price American options, Longstaff and Schwartz (2001) combined Monte Carlo

Simulation with the Least Squares method, a regression-based method, by using back-

wards induction to approximate conditional continuation values. The methods of Tsit-

siklis and Van Roy (2001) and Carriere (1996) are similar to least-squares Monte Carlo

(LSM) algorithm of Longstaff and Schwartz (2001). Clement et al. (2001) analyzed

the convergence of LSM. Many of these methods are explained in Glasserman (2003)

and Tavella (2002). Throughout this thesis, LSM refers to the algorithm of Longstaff

and Schwartz.

Primal-dual representations of the American option problem have been estab-

lished by Haugh and Kogan (2004), Rogers (2002) and Andersen and Broadie (2004).

Duality approach allows both an upper and lower bound to be calculated through a

minimization problem over a class of martingales or super martingales. These methods

are also described in Glasserman (2003).

FD and tree techniques are traditional methods for pricing (Longstaff and Schwartz,

2001). Most of these methods try to approximate the continuation values by con-

sidering the problem in a stochastic dynamic programming framework. For pricing

American-style securities, they are efficient in the one-dimensional case whereas simu-

lation methods work well for multi-asset derivatives (Broadie et al., 2000).

16

4.2. Simulation of American Options

For the simulation of American options, there are a number of basic issues to be

explained before we can discuss implementation.

It is emphasized that Bermudan options can be positioned between American and

European options. In simulation, due to the inability to simulate in continuous time,

an American option is priced under the assumption that it has Bermudan style, and

thus only discrete exercise opportunities exist. So, if an American option is exercisable

at any time t where 0 ≤ t ≤ T , we restrict the option such that it can be exercised

only at a fixed set of exercise opportunities 0 < t1 < t2 < . . . < td , where d is the

number of exercisable time steps.

In addition, the American option holder compares the payoff from early exer-

cise and expected continuation value to make decision to keep the option alive or not,

because the rational behavior for the holder is to maximize the income by this com-

parison. The payoff function is determined according to the option type. Hence, the

key insight in simulation for pricing American options is to estimate the continuation

value.

Moreover, the Exercise Boundary is an essential term in pricing American options.

It refers to a threshold value for each time step which shows if we should exercise or

not. For a put, we should exercise when the price of the underlying is below the exercise

boundary while it is opposite if the option is a call. The boundary for expiration date

is exactly the strike price. Approximation methods deal with the approximation of the

exercise boundary. Simulation techniques try to price the option by determining the

stopping time for each path after generating paths and estimating the continuation

value. However, in a sense, they look for the exercise boundary in the lifetime of the

option. Thus, since we have an idea about threshold values for early exercise, which

refer to exercise boundary at any exercisable time step, the simulation of the price will

be simple and fast.

17

Furthermore, it is important to understand the biases in simulation for pricing

American options. Most of the methods terminate simulated paths according to their

approximation of the optimal exercise boundary (Glasserman, 2003). This can lead

to both low and high biases. Using a sub-optimal exercise policy introduces low bias.

However, the reason behind high bias is the usage of future information in paths while

making the exercise decision.

4.3. LSM algorithm of Longstaff and Schwartz

At any exercise time, the immediate cash flow generated by early exercise and

expected payoff from continuation is compared by the American option holder. Ac-

cording to this comparison, the decision to exercise the option or to keep it alive is

made. Longstaff and Schwartz use regression to estimate the continuation value.

Longstaff and Schwartz (2001) combine Monte Carlo Simulation with the Least

Squares method by using backwards induction to approximate conditional continuation

values to price American options. The main idea in their least-squares Monte Carlo

(LSM) approach is the usage of least squares to estimate conditional continuation

values to decide to exercise the option or not. In the algorithm, the American option is

assumed to be of Bermudan style, which has discrete time steps allowing early exercise.

After generating paths of the underlying for discrete time steps, the cash flows from

early exercise and continuation values are compared to decide to exercise or to continue

holding the option at each time step by backward induction. The continuation values

at an exercise date are estimated by regressing the discounted subsequent realized

cash flows from continuation on a set of basis functions. The regression fitted at each

exercisable time step includes only the paths where the option is in the money and

results are used only for the comparison. The price of the option is computed by

the means of discounted cash flows according to the stopping rule that the algorithm

found.1

Let a Markov process {X(t), 0 ≤ t ≤ T} denote all necessary information about

1See the numerical example in Longstaff and Schwartz (2001) for a better understanding

18

the current price of the underlying asset. To reduce notation, X (ti) will be written

as Xi, which is the state of the underlying Markov process at the ith time step, where

i = 1, . . . , d. The discrete-time process X0 = X(0), X1, . . . , Xd is a Markov chain.

Vi(x) and hi(x) denote the value of the option and the payoff function at time ti given

Xi = x . The continuation value in state x at date ti is

Ci (x) = E [Vi+1 (Xi+1) |Xi = x] =M∑r=1

βirψr(x)

for some basis functions ψr and coefficients βir, r = 1, . . . ,M . The coefficients could

be estimated by pairs (Xi, Vi+1(Xi+1)). Vi+1 is unknown in practice. Hence, it must be

replaced by the discounted estimated values Vi+1 at downstream nodes. So, regression

coefficients define the estimate of the continuation value at an arbitrary point x:

Ci (x) = βTi ψ(x) (4.1)

with

βTi =(βi1, ...βiM

), ψ(x) = (ψ1(x), ..., ψM(x))T

Then, the algorithm can be summarized as follows:

i. Simulate n independent paths of the Markov chain {X0, X1, . . . , Xd} for d time

steps,

ii. At terminal nodes, set Vdj = hd (Xdj) , j = 1, . . . , n,

iii. Apply backward induction: for i = d− 1, . . . , 1,

• Specify the set I, which is composed of paths that are in the money

• Discount the values Vi+1,j, j ∈ I to be input for regression

• Given estimated values Vi+1,j, j ∈ I, use regression to calculate βi,

• Estimate continuation values by using Equation 4.1,

• If hi (Xij) ≥ Ci (Xij), set Vij = hi (Xij), else set Vij = Vi+1,j, j ∈ I ,

iv. Calculate(V11 + ...+ V1n

)/n and discount it to find V0

19

From the perspective of exercise boundary, we know that at the expiration date,

the exact boundary is the strike price K. If the option is put, then optimal stopping

time τ ∗ has the form

τ ∗ = inf{t ≥: S(t) ≤ b∗(t)}

for some optimal exercise boundary b∗(t), where S(t) is the price of the underlying at

time t.

Diagnostic of the LSM algorithm has been tested by applying the regression pa-

rameters obtained by one set of paths to another set of paths. So, price estimates

calculated by using different two sets of paths and same regression functions are ap-

proximately similar and accuracy of the approach is supported.

There are several essential issues considered in LSM algorithm of Longstaff and

Schwartz. First, only in-the-money paths, where the payoff function is positive, are

selected as input to the regression. In other words, paths that do not create any cash

flow when exercised at that time step are neglected in regression. In addition, all

values are normalized by dividing with the strike price and the conditional expectation

function is estimated in the normalized space for American put illustration to avoid

computational underflows while using a set of weighted Laguerre polynomials:

L0 = exp(−X/2) (4.2)

L1 = exp(−X/2)(1−X) (4.3)

L2 = exp(−X/2)(1− 2X +X2/2) (4.4)

Ln = exp(−X/2)eX

n!

dn

dXn(Xne−X) (4.5)

This process does not affect the option value since the unnormalized values are later

transformed back.

LSM method produces a low-biased estimator because for each path, an estimate

20

of optimal exercise time is used (Lemieux and La, 2005). Besides being versatile

and easy to implement, the approach is easily adjustable to different cases such as

exotic American options and American options with jump diffusions as well as multi-

dimensional American options (Laprise et al., 2006). However, Longstaff and Schwartz

(2001) claim that the results of LSM are robust with respect to the choice of basis

functions. For example, they assert that to price a one-dimensional Amrican put

option, it is sufficient to use a constant and three basis functions, which are the first

three weighted Laguerre polynomials, as in equations 4.2-4.4. Moreover, as a simulation

technique, it is possible to use LSM for parallel computing. In other words, 50000 paths

might be generated in one CPU or 5000 paths each on 10 CPUs.

Meanwhile, several variance reduction techniques were suggested for pricing Amer-

ican options via the LSM Algorithm (Bolia et al., 2004; Ehrlichman and Henderson,

2006; Lemieux and La, 2005)

21

5. ASSESSMENT OF LSM ALGORITHM

The LSM algorithm of Longstaff and Schwartz was explained in the previous

chapter. In this chapter, information about how to implement the algorithm in R and

results of experiments made by using different basis functions are provided. Then,

some problems in the algorithm and ways to improve them are discussed.

5.1. Implementation of LSM in R

R is a free software environment for statistical computing and graphics. Through-

out this thesis, R version 2.7.0 has been used. No package in R exists for the LSM

algorithm. You can find R codes for the implementation of the LSM algorithm in

Appendix A.2. In this section, we simply explain our codes.

Before explaining our implementation, an issue should be cleared. The stock

prices for all paths for each time step can be generated by geometric Brownian motion,

which is decribed in Section 3.1, through Equation 3.8. However, in this case, a matrix

is needed to store the stock prices at all time steps because of the working backwards

principle in LSM. The Brownian-Bridge offers a solution to decrease memory require-

ments (Glasserman, 2003). It allows us to interpolate a value of a Brownian motion

given two endpoints. Thus, for any time t, such that 0 < t < T , where T is the expi-

ration date, St can be calculated by a Brownian Bridge construction if S0 and ST are

known. Hence, it is possible to start with simulating the prices ST at maturity and

continue by simulating St. The expectation and the variance can be calculated by the

simple formulae:

E(logSt) = log Sa +t− ab− a

(logSb − logSa) (5.1)

V (logSt) = σ2 (b− t) (t− a)

b− a(5.2)

where a < t < b. So, at each time step, storing two arrays with length of n, the

22

number of paths generated, is enough instead of storing a n x d matrix. Hence, to

reduce memory requirements and to be able to simulate a large number of paths, the

Brownian Bridge idea is used in our implementation.

The function for pricing is named simul US LSM. The initialization stage is com-

posed of the normalization of all prices and cash flows by dividing the starting stock

price over the exercise price K and finding the time interval throughout simulation.

The first stage, which takes place at the expiration date, includes simulating the

stock prices and calculating the cash flows. For this purpose, the stock prices at T

are simulated using the Equation 3.8. For the version, where half of the stock prices

are simulated by using antithetic variates, generated normal variates are needed to be

stored. In this case, all stock prices to apply regression are combined in one array such

that the first half is generated from normal variates and the second uses antithetic

variates.

In the next stage, a for loop from d − 1 to 1 is executed to deal with the calcu-

lations in each time step. In this for loop first the stock prices at that time step for

all paths are generated by using Brownian Bridge through equations 5.1 and 5.2. By

the payoff function, in-the-money paths are selected to be considered in the regression.

For this purpose, an array named idx that stores the indices of paths that are in the

money is assigned. This facilitates carrying out the operations. Before fitting regres-

sion, future cash flows are discounted with the risk-free interest rate to be input to the

regression. Basis functions for the regression are defined as constant and the first three

Laguerre polynomials, which can be seen in equations 4.2-4.4, as indicated in the paper

of Longstaff and Schwartz (2001). The coefficients that are calculated by regression

enable us to estimate the continuation values for selected paths. The estimated con-

tinuation values are saved in a different array. According to the comparison between

estimated continuation values and the payoff function, ones and zeros are assigned to

exercise flag EF.

At this step, the continuation value for the current time step is composed of

23

two parts; one coming from early exercise for the paths where EF contains 1s and

continuation values for future time step for the paths where EF contains 0s. The first

part might be calculated in two ways; by exercising the paths when EF is equal to 1

or by selecting the maximum stock price where EF is 1 so that below that price there

will be early exercise. After computing the value at the current time step, the stock

prices at ti are assigned to stock prices at time ti+1 to continue with the previous time

step.

The for loop ends when it completes the calculations for t1. The remaining part

is the discount process of the last continuation value to t0, which represents the initial

date. The mean of all paths after discounting gives us the American option price.

It is essential to claim that in the case that antithetic variates are used, the average

of first half and second half should be taken into consideration before computing the

mean. The function prints out the American option price, the European option price (if

the option would be European style), the early exercise premium (the value difference

between American and European style) and the standard error of estimation.

5.2. How many basis functions are necessary?

The algorithm is run by varying the initial stock price, the volatility and the

lifetime. In Section 4.3, it was stated that Longstaff and Schwartz (2001) assert con-

sidering three Laguerre polynomials is sufficient. Therefore, experiments were carried

out by changing the number of Laguerre polynomials used as basis functions in algo-

rithm, according to the general equation of Laguerre polynomials 4.5. Each case was

replicated for 100 times.

In Table 5.1, besides the results of these experiments, the prices from the article

of Longstaff and Schwartz (2001), calculated by Finite Difference (FD) method and

LSM can be seen. Lk denotes first k number of Laguerre polynomials selected as basis

functions. There is not too much difference between the results of FD, LSM and the

cases where four and five Laguerre polynomials are used. This is a fact which suits to

the statement of Longstaff and Schwartz (2001); only few basis functions are needed

24

Table 5.1. Price estimates for different number of basis functions and varied initial

stock price, volatility and lifetime

L1 L2 L3 L4 L5 LSM FD

S0 = 38 sigma=0.2 T=1 3.213 3.236 3.246 3.247 3.247 3.244 3.25

T=2 3.691 3.727 3.733 3.738 3.738 3.735 3.745

sigma=0.4 T=1 6.116 6.136 6.141 6.141 6.142 6.139 6.148

T=2 7.613 7.649 7.668 7.667 7.671 7.669 7.67

S0 = 40 sigma=0.2 T=1 2.289 2.307 2.312 2.315 2.315 2.313 2.314

T=2 2.844 2.872 2.881 2.886 2.884 2.879 2.885

sigma=0.4 T=1 5.283 5.299 5.312 5.321 5.315 5.308 5.312

T=2 6.869 6.908 6.918 6.920 6.919 6.921 6.92

S0 = 42 sigma=0.2 T=1 1.601 1.613 1.616 1.616 1.617 1.617 1.617

T=2 2.18 2.2 2.208 2.208 2.210 2.206 2.212

sigma=0.4 T=1 4.563 4.574 4.584 4.588 4.587 4.588 4.582

T=2 6.204 6.232 6.240 6.243 6.245 6.243 6.248

to approximate conditional expectation closely and adding more basis functions has

little effect on the value of the option. However, it is obvious that when less than three

polynomials are used; price estimates are not very close to the ones of FD.

5.3. Problems in LSM algorithm

In this section, problems in the LSM are discussed. After explaining the influence

of the regression on the speed, its accuracy in estimating the early exercise boundary

will be checked.

5.3.1. Inefficiency of Regression

It is asserted that continuation values at a time step are estimated through a re-

gression to decide whether to exercise or not. The comparison between these estimated

values and immediate cash flows will give us a threshold value, which is the maximum

stock price for which early exercise takes place. Thus, it is the value where estimated

25

continuation values and immediate cash flows from early exercise intersect each other.

In Figure 5.1, an example of this case taken from a simulation at time td−1, where d is

the number of exercise opportunities, is shown. Notice that the figure is just the part of

the whole plot that is close to the intersection point. By fitting the regression to many

stock prices that are far from threshold value, LSM algorithm estimates continuation

values for stock prices that we are not interested in because it takes all in-the money

points as input to regression to find the threshold value at a time step.

Figure 5.1. Plot of LSM estimates of continuation values versus immediate cash flows

from early exercise at time td−1

Fitting a regression takes significant time. When basis functions are Laguerre

polynomials, it is obviously longer than usual. In other words, while there is the

26

interesting region where the two functions intersect each other as in Figure 5.1, the

algorithm includes all stock prices that are smaller than the strike price K, instead of

focusing on the important region. This leads to inefficiency in computational time and

relatively high error in estimation.

It is obvious that the expected continuation values are only estimated to be able

to find an exercise boundary, which has influence on the price. However, all generated

paths are not considered. Therefore, if a better way to specify the exercise threshold

could be found, we would not need to estimate continuation values and would price

the option according to the boundary by directly using the stock prices.

5.3.2. Boundary Estimation

Let us suppose there are d time steps that option can be exercised in (0, T ], where

td = T . At time td−1, it is possible to find the exact early exercise boundary by using

the Black-Scholes (BS) formula and the Newton-Raphson (NR) method. In Section

3.1, it is indicated that the BS formula gives the exact value of the European call or

put option. Also, it is mentioned that in the LSM algorithm, an American option is

priced under the assumption that it is Bermudan. Hence, since time td−1, the option

is a European option that has life time in (td−1 , td]. It is possible to specify an exact

boundary at time td−1, such that there must be a stock price that makes the value

of the European option computed from BS formula, which is also the continuation

value of the American option, equal to the immediate income from early exercise of

American option at td−1. In other words, at time td−1, the option does not differ from

a European style. So, we can use the price of the European option starting at td−1 and

with expiry td = T as the exact continuation value for the American option at td−1.

Since we know the threshold is the stock price which equalizes the continuation value

to the early exercise value, it can be obtained by equating the price of the European

option to the payoff at td−1. Hence, by the help of BS formula, the threshold for a put

is the stock price that satisfies the equation:

V EUp − (K − S)+ = 0

27

Table 5.2. Early exercise boundary calculated by LSM algorithm for a Bermudan

option with two exercise opportunities and comparison with exact values

Time of 1st Exercise

11/12 10/12 9/12 8/12 7/12 6/12

L1 36.3309 36.06 35.9347 35.8921 35.8621 35.8946

L2 36.536 36.4974 36.4543 36.4342 36.4301 36.4303

L3 37.1316 36.9665 36.8417 36.7319 36.6238 36.5559

L4 37.4576 37.145 36.9341 36.7636 36.6455 36.5483

L5 37.6021 37.1821 36.9311 36.767 36.642 36.5545

BS+NR 37.6472 37.1941 36.9366 36.7663 36.6457 36.5571

This means that for a specific value of S, which is called early exercise boundary, the

continuation value equals the immediate income gained by early exercise. If V EUp is

replaced by its value of the BS formula, equation becomes:

Ke−rTN(−d2)− SN(−d1)− (K − S)+ = 0 (5.3)

This equation can be solved by the NR method. A root of f(x) = 0 can be found with

an initial guess of x0

xk+1 = xk −f (xk)

f ′(xk)

(Acton, 1990). The algorithm might be stopped when

|xk+1 − xk||xk|

≤ ε

where ε is the acceptable error and can be determined by the user. Since f(x) is the

left hand side of Equation 5.3 and x is the asset price S, the root found by the NR

method is the element of exercise boundary for td−1. The R implementation of the

algorithm can be seen in Appendix A.3.

28

To illustrate the idea, let us consider a Bermudan put option on a share of a non-

dividend paying stock that has initial value 40 and volatility of 20 percent. The option

is exercisable at an exercise price of 40 and has the maturity of one year. The risk-free

interest rate is six per cent. For the first example, let us assume that the number of

exercisable time steps d is two; one is at maturity and the other one varies. In other

words, there is only a single early exercise opportunity. At time t2 = T , the exact

boundary is the exercise price, so, the option will be exercised if it is in the money.

Thus, the exact boundary at maturity is the strike price. We experimented to find the

early exercise threshold value by comparing LSM algorithm and a combination of BS

formula and NR method. In the experiments where the early exercise boundary found

by BS formula and NR method, ε is taken as 10−8. Results are summarized in Table 5.2

by varying the first exercisable time and the number of Laguerre polynomials, where

Lk denotes first k weighted Laguerre polynomials selected as basis functions. Since

the threshold values found by BS formula and NR method are the exact boundaries,

it can be seen that the performance of LSM decreases when the exercisable time step

is close to maturity. In addition, the boundaries computed by LSM are always smaller

than the exact boundaries and get closer to the exact ones as we increase the number

of basis functions used in the algorithm, however, they are still not identical for the

situations in which the early exercise is allowed close to the maturity. This property

is a problem for the algorithm because for pricing American options we use Bermudan

options with many exercise opportunities.

Altogether, the LSM algorithm has two main problems; the computational load

of fitting the regression and the inability to estimate the exercise boundary.

5.4. Improvement of Region of Regression

In Section 5.3.1, the problem of the LSM algorithm with regression, which is the

fact including all in-the-money points instead of focusing on the important region is de-

scribed. In this section, we propose a way to improve this situation, its implementation

in R and numerical results.

29

5.4.1. The Idea

Longstaff and Schwartz use Laguerre polynomials as basis functions in regression.

It takes considerable time to fit such a regression to all in-the-money paths. However, if

we had an idea about the early exercise threshold, it would be possible to fit a simpler

linear regression around that value and modify that pre-estimated threshold value.

One possibility to limit the region where regression is applied at a time step is the

usage of the exercise boundary at the subsequent time step. The other, which we will

focus on, is the usage of the LSM algorithm. Using a relatively small number of paths

to estimate the threshold values in LSM algorithm might give an idea about where

the thresholds are. After that, those values might be improved by applying simple

regressions around that region. Thus, we propose that the LSM algorithm might be

enhanced by following these steps;

• First run the LSM algorithm with a specific small number of paths selected from

generated ones, to have an idea where the early exercise threshold is.

• At each time step specify a distance, such that, a specific number of paths exists

in that region close to the pre-estimated threshold.

• Instead of Laguerre polynomials, consider a simpler basis function, such as stock

price at that time step, fit regression and find the modified threshold value.

• Value the option according to the modified threshold values.

By this algorithm, it is aimed to reach a speed-up while getting identical results with

the LSM algorithm and the FD method. Throughout this thesis, this method will be

called Simple Regression Approach (SRA).

In the pre-estimating stage, we have two choices in selecting paths to be consid-

ered; random selection from generated paths or selection of paths by beginning from

the first generated one. Since, all paths are randomly generated, we do not think that

it will cause a problem in bias to select the first samples. In experiments, the second

approach is used. For some time steps the LSM algorithm might have problems to find

a threshold value. In these cases, it is assumed the threshold value at time ti is equal

30

to the one at time ti+1 if it cannot be computed by the algorithm.

After commenting how to implement SRA in R, the algorithm will be tested by

a number of experiments.

5.4.2. R Implementation of SRA

The implementation of this algorithm in R can be seen in Appendix A.4. The

implementation is not much different from the one of LSM. The difference in the

function simul US SRA is the stage that estimates the threshold value with small

number of paths and fits a simple regression to the values closer to the threshold value

than a specified distance.

Two new arguments were added to the definition of the previous function; npilot,

as the percentage of generated paths for the pilot study with the LSM algorithm and

nreg, as the percentage of generated paths to be considered in the simple regression.

There is no difference in commands before the for loop except identifying an array for

thresholds to save. Since Brownian Bridge is used, it is impossible to have an idea about

all the threshold values at each time step at once. Hence, we add this fitting regression

with a small number of paths at the beginning of the for loop, after generating the stock

prices at that time step. We store that threshold value and specify a distance according

to the input value of nreg. For this purpose, we subtract the threshold value from all

asset prices at that time step, take their absolute value and store them in an array.

The nregth element of the sorted array is our distance. After that, regression is fitted

by using simpler basis functions, a constant, the asset price, and a new threshold value

are obtained. Before going to the previous time step, the exercise flag EF is assigned

by the threshold value and the value of the option at that time step is calculated. In

the end of the for loop, again we change the current asset price as the subsequent

asset price as we use the Brownian Bridge. The function prints American option price,

standard error, European option price and early exercise premium.

In addition, the computational time can be taken by using the function sys-

31

tem.time.

5.4.3. Numerical Results

Let us consider the example that we used to show the problem of LSM in estimat-

ing the exercise boundary. The exercise boundary calculated by SRA for that example

is compared with the exact values obtained by the BS formula and NR method in

Table 5.3. npilot refers to the percent of all paths to be considered for the pilot study

with the LSM algorithm whereas nreg refers to the percentage of all paths used for

simple regression. In contrast to the LSM algorithm, SRA estimates the boundaries

quite well, even when the time interval gets smaller.

Now, the algorithm is run to price the American option and the results are com-

pared with FD and LSM results given in the paper of Longstaff and Schwartz (2001).

Since the boundaries are estimated well when npilot is 1 percent and nreg is 10 per-

cent, those values were selected for further experiments, to reduce the execution time.

However, in the cases where the percentage of in-the-money paths is less due to the

initial asset price, a higher percentage value would be better to select for npilot.

Table 5.4 summarizes the results for the American put example considered pre-

viously. According to the table, SRA gives sensible results closer to FD results than

the LSM algorithm. Also, time reduction factors (TRFs), which means the fraction

of time used by LSM algorithm over time used by simple regression approach, are re-

ported and show that the computational effort is decreased. TRFs are higher when

the option begins in-the-money, in other words, when the initial asset price is lower

than the strike price. As the life of the option gets longer, TRFs tends to decrease,

however, it is not valid for the cases in which the initial asset price is higher than the

strike price. Furthermore, it can be said that increased volatility of the asset leads to

higher TRFs.

Altogether, SRA is an alternative simulation method that can be used for pric-

ing one-dimensional American options with price estimates that are similar to those

32

Table 5.3. Early exercise boundary calculated by SRA for a Bermudan option with

two exercise opportunities and comparison with exact values

Time of 1st Exercise

11/12 10/12 9/12 8/12 7/12 6/12

npilot=10% nreg=10% 37.61998 37.16731 36.88797 36.74795 36.63377 36.52508

nreg=5% 37.54787 37.18704 36.91227 36.73801 36.62277 36.55265

npilot=5% nreg=10% 37.60546 37.15817 36.87599 36.74097 36.62698 36.53988

nreg=5% 37.54228 37.17075 36.91736 36.73636 36.61452 36.55383

npilot=1% nreg=10% 37.60229 37.17075 36.89427 36.75314 36.63698 36.57485

nreg=5% 37.51084 37.14335 36.87268 36.72641 36.6116 36.51356

BS+NR 37.6472 37.1941 36.9366 36.7663 36.6457 36.5571

calculated by FD and with smaller computational costs than LSM.

Table 5.4. Comparison of prices calculated by SRA, LSM and FD, and TRFs varying

initial stock price, volatility and lifetime

LSM FD SRA TRF

S0=38 sigma=0.2 T=1 3.244 3.25 3.249 4.61

T=2 3.735 3.745 3.744 4.03

sigma=0.4 T=1 6.139 6.148 6.147 4.19

T=2 7.669 7.67 7.672 3.93

S0=40 sigma=0.2 T=1 2.313 2.314 2.314 3.06

T=2 2.879 2.885 2.886 2.66

sigma=0.4 T=1 5.308 5.312 5.313 3.45

T=2 6.921 6.92 6.916 3.32

S0=42 sigma=0.2 T=1 1.617 1.617 1.616 1.77

T=2 2.206 2.212 2.21 1.85

sigma=0.4 T=1 4.588 4.582 4.584 2.77

T=2 6.243 6.248 6.241 2.87

33

6. OPTIMIZATION APPROACH

An American option holder tries to maximize his income by making a comparison

between two choices. From this point of view, optimization might be a useful idea.

The approach used in this chapter is the optimization of the exercise boundary by

backwards induction. In other words, it is similar to the LSM algorithm with respect to

the working principle but searches the optimum threshold value for the maximization

of the total value instead of fitting regression. In addition, the optimization approach

directly uses future information in pricing whereas in regression based methods, future

information is only used for the regression to estimate future cash flows. As stated in

Section 4.2, this fact causes the optimization approach to have high bias, in contrast

to the low bias of regression based methods.

6.1. R Implementation of Optimization Approach

R codes for the implementation of the optimization approach can be seen in

Appendix A.5. Since we do not use LSM algorithm and Laguerre polynomials, there

is no need to normalize the prices and cash flows with the strike price. Then, the

initialization stage of the function simul US OPT is only composed of calculating the

length of the time interval, generating stock prices at maturity and defining an array

to save threshold values. There are also a few changes inside the codes, especially in

the for loop.

The content of the for loop is different from the one used in LSM or SRA. At

each time step before the maturity date, the order of the array which consists of stock

prices is stored. By using the same indices for this order, continuation values and early

exercise incomes are written into two arrays with length of n, equal to the number of

paths generated. Notice that if antithetic variates are used, each will have a length of

2n. Next, the total value array is formed for each stock price, which is a candidate for

the boundary value, according to the fact that total value of a put option is the sum of

34

cumulative sum of early exercise incomes equal to or below the threshold candidate and

cumulative sum of continuation values above the threshold candidate. The threshold

value is selected as the stock price that maximizes the total value by using the function

which.max. Therefore, early exercise for put options takes place for the stock prices

under the threshold values and estimated continuation values are calculated according

to this exercise policy before performing the operations for the next time step in the

loop. Stock prices at time ti are assigned as future stock prices for time ti−1.

After the calculations within the for loop, the function calculates the average

of discounted expected cash flows of all paths. If antithetic variates are used, before

taking the mean, the cash flows of the paths generated by normal variates and the cash

flows of paths generated by antithetic variates must be averaged. As an output, the

price of the American option and the standard error of the simulation are printed.

6.2. Numerical Results

Similar to the experiments done previously, the accuracy of the optimization

approach is tested by pricing the American put option of the example used in the

paper of Longstaff and Schwartz (2001). It is possible to compare the results of different

methods for a variety of cases with the help of Table 6.1. In the optimization approach,

for all cases, price estimates achieved by applying antithetic variates are significantly

higher than the estimates of LSM, SRA and FD. The main reason of this difference

seems to be the usage of future information.

Besides being high biased, there is another issue about this approach, which might

also lead to inefficient results. Figure 6.1 shows a plot of total values versus stock prices

at time td−1, where d is the number of time steps and td = T is the expiration date.

As seen in Figure 6.1, several other stock prices, when considered as a threshold,

conclude in total values that are close to the maximum. However, in this approach,

the stock price that leads to maximum total value is taken as the threshold value at

that time step. This might affect the choice of the threshold value in the previous

exercise opportunity date. In other words, maximizing the total value for all paths at

35

Table 6.1. Prices calculated by optimization approach, LSM and FD, varying initial

stock price, volatility and lifetime

LSM FD OPT s.e

S0=38 sigma=0.2 T=1 3.244 3.25 3.263 0.005

T=2 3.735 3.745 3.76 0.006

sigma=0.4 T=1 6.139 6.148 6.171 0.008

T=2 7.669 7.67 7.701 0.01

S0=40 sigma=0.2 T=1 2.313 2.314 2.326 0.005

T=2 2.879 2.885 2.9 0.006

sigma=0.4 T=1 5.308 5.312 5.334 0.009

T=2 6.921 6.92 6.948 0.01

S0=42 sigma=0.2 T=1 1.617 1.617 1.626 0.006

T=2 2.206 2.212 2.228 0.007

sigma=0.4 T=1 4.588 4.582 4.603 0.01

T=2 6.243 6.248 6.276 0.011

each time step does not necessarily lead to the maximization of the total value for the

entire simulation in the whole lifetime of the option. In spite of this, the estimates

for the option price are biased significantly high. Nevertheless, this approach might be

used to obtain an upper bound for the option price. Additionally, the threshold value

estimates of this approach might be examined by using the two-step Bermudan case.

Altogether, the optimization approach is run to price American put option. How-

ever, the results are not identical to the ones calculated by using LSM and FD but

too high. So, an upper bound for the option price might be obtained by using the

optimization approach.

36

Figure 6.1. Plot of total values for different early exercise thresholds at time td−1

37

7. MULTI-ASSET OPTIONS

The rapid pace of innovations in financial markets has resulted in an increasing

interest in options written on multiple assets (Detemple et al., 2003).

According to Das (2004), multi-asset options form a subgroup of exotic options.

Multi-Asset options are exotic options where the payoff depends on two or more assets.

Thus, the correlation between the underlying asset prices must be considered as a factor

in the behavior of the option. Options on the maximum of several assets, spread and

basket options are examples of multi-asset options.

The primary reason for the use of multi-asset options is to hedge or trade specific

types of relative asset price movements (Das, 2004). Variation in risk profile and

reduction of option premium are other possible reasons for using multi-asset options.

Before giving a brief summary about the general framework and the formulation

of multi-asset options, we define and explain two option types that will be priced in

further sections; spread options and options on the maximum of k assets, which are

classified as rainbow options by Das (2004).

7.1. Types of Multi-Asset Options

In this section, we briefly explain the spread options and option on the maximum

of k assets. In addition, the formulae to calculate the exact price of a European option

on the maximum of two assets and the formulae of Kirk’s approximation, which is an

approximation method for European-style spread options, are given.

7.1.1. Options on the Maximum of k Assets

Options on maximum of k assets, in other words maximum options or best-of

options, are options whose payoff depends on the difference between the maximum

38

price of k assets and the strike price (Detemple et al., 2003). Let us assume there is

an option on the maximum of two assets. Then, its payoff function for a call is

hc(S1, S2

)= max

(max

(S1, S2

)−K, 0

)and for a put, the function becomes

hp(S1, S2

)= max

(K −max

(S1, S2

), 0)

where S1 and S2 represent the prices of assets, K denotes the exercise price and Q1

and Q2 are the quantities of the asset. In practice, generally quantities are assumed as

Q1 = Q2 = 1.

According to Detemple et al. (2003), an option on the maximum of several

assets is one of the fundamental building blocks of complex financial products and

capital budgeting operations. Das (2004) asserts that asset managers generally use

this type of options. Commonly they are written on baskets of equity stocks or equity

indexes as well as on the performance of different asset classes such as equity and fixed

income/bonds.

7.1.1.1. Exact Price of a European call on the maximum of two assets. The price of

a European option on the maximum of k assets has a closed form solution only when

the number of underlying assets is two. Haug (2007) gives the formulae to find the

exact value of a European option on the maximum of two assets in Chapter 5. Due to

the case that will be analyzed, only the formulation for the price of a call is given in

this section. The price of a European call option on the maximum of two assets is:

V Maximumc = S1

0e(b1−r)TM (y1, d; ρ1) + S2

0e(b2−r)TM

(y2,−d+ σ

√T ; ρ2

)−Ke−rT

[1−M

(−y1 + σ1

√T ,−y2 + σ2

√T ; ρ

)](7.1)

39

where

σ =√σ2

1 + σ22 − 2ρσ1σ2 (7.2)

d =ln (S1

0/S20) + (b1 − b2 + σ2/2)T

σ√T

(7.3)

y1 =ln (S1

0/K) + (b1 + σ21/2)T

σ1

√T

(7.4)

y2 =ln (S2

0/K) + (b2 + σ22/2)T

σ2

√T

(7.5)

ρ1 =σ1 − ρσ2

σ(7.6)

ρ2 =σ2 − ρσ1

σ(7.7)

and M (., .; .) is the cumulative distribution function of bivariate standard normal dis-

tribution. Si0, bi, σi are respectively initial price, cost-of-carry and volatility for the

asset i, where i = 1, 2. R codes that calculate the exact price of a European call option

on the maximum of two assets according to the formulas can be seen in Appendix A.6.

7.1.2. Spread Options

A spread option is an option type whose value depends on the difference between

the prices of two or more assets (Haug, 2007). The payoff of a call spread option on

two assets is:

hc(S1, S2

)= max

((Q1S

1 −Q2S2)−K, 0

)(7.8)

where S1 and S2 are the asset prices. Notice that if the spread option is a put, then

the payoff function is:

hp(S1, S2

)= max

(K −

(Q1S

1 −Q2S2), 0)

Das (2004) asserts that spread options can be used to capture price differentials

between commodities that are related in two ways; demand substitution or transfor-

40

mation potential. Demand substitution refers to the substitution between alternative

investments such as bonds or equity indexes. Transformation potential is the term for

assets where one asset is a price input into the other.

Underlying assets can be any type of financial products such as equities, bonds

and currencies (Haug, 2007). The commodity spreads that are based on the difference

between the inputs and outputs of a process enable the trader to gain exposure to the

production process of the commodity (Investopedia, 2008). Crack and spark spreads

are common instances for this type of options.

Spread options are generally traded in the over-the-counter market. American

spread options on oil products are traded in the New York Mercantile Exchange (Haug,

2007). For instance, the heating oil crack is a spread option on the heating oil and crude

oil whereas the gasoline crack is a spread option on unleaded gasoline and crude oil.

These types of options are used for hedging purposes by oil refineries. In addition, Das

(2004) mentions that interest rate spread options are widely used in the US market.

He adds that another area of significant activity is the practice of cross-market spread

options between the major reserve currencies such as US dollars, Euro and Yen.

7.1.2.1. Kirk’s approximation for Spread Options. Kirk (1995) approximated the price

of a European spread option by using the Black-Scholes model by a set of transforma-

tion. According to Kirk’s approximation, the value of a call V spreadc or the value of a

put V spreadp can be calculated by:

V spreadc ≈

(Q2S

20e

(b2−r)T +Ke−rT)

[SN (d1)−N (d2)] (7.9)

V spreadp ≈

(Q2S

20e

(b2−r)T +Ke−rT)

[N (−d2)− SN (−d1)] (7.10)

41

where

d1 =ln (S) + (σ2/2)T

σ√T

, d2 = d1− σ√T (7.11)

S =Q1S

10e

(b1−r)T

Q2S20e

(b2−r)T +Ke−rT(7.12)

and the volatility can be approximated by

σ ≈√σ2

1 + (σ2F )2 − 2ρσ1σ2F (7.13)

where

F =Q2S

20e

(b2−r)T

Q2S20e

(b2−r)T +Ke−rT(7.14)

and

S10 : Initial Price of Asset One

S20 : Initial Price of Asset Two

Q1 : Quantity of Asset One

Q2 : Quantity of Asset Two

K : Strike Price

T : Time to expiration

b1 : Cost-of-carry of Asset One

b2 : Cost-of-carry of Asset Two

r : Risk-free interest rate

σ1 : Volatility of Asset One

σ2 : Volatility of Asset One

ρ : Correlation

R codes of the Kirk’s approximation can be seen in Appendix A.7.

42

7.2. Simulation of multi-asset options

While simulating multi-asset options, it is obvious that we need to simulate prices

of a number of assets according to the dimension. Equation 3.1 is a SDE for the

one-dimensional case. For multi asset options, prices of assets at a time t Sit , where

i = 1, . . . , k and k is the number of assets, SDE becomes:

dSit = µiSitdt+ σiS

itW

it (7.15)

In the examples considered for the multi-dimensional case, generally assets are dividend-

paying. Hence, the continuous dividend yield should be included in the drift µ under

the risk neutral measure (Wilmott, 2006). Thus, the drift should be replaced by the

difference of risk-free interest rate r and continuous dividend yield δ. The Equation

7.15 becomes

dSit = (r − δi)Sitdt+ σiSitW

it (7.16)

Then, the prices of the assets at a time t can be simulated by the following equation:

Sit = Si0 exp

((r − δi −

1

2σ2i

)t+ σi

√tZi

)

Notice that processes Zi are correlated due to the correlation between asset prices.

Cholesky decomposition helps us to transform the uncorrelated geometric Brownian

motions (GBMs) into the correlated ones. It is explained in detail in the next sub-

section.

7.2.1. Cholesky decomposition

According to Cholesky factorization, a symmetric positive-definite matrix A can

be decomposed into a lower triangular matrix L and the transpose of the lower trian-

43

gular matrix L:

A = L.LT

In Monte Carlo simulation, the Cholesky decomposition enables us to generate

correlated variates with a transformation of uncorrelated variates (Wilmott, 2006).

Let us suppose we are given a correlation matrix A and we have generated matrix

Z = (Z1, . . . , Zk)′ of k independent standard normal variates. Then, correlated variates

Zc can be obtained by the transformation:

Zc = L.Z

where L is the Cholesky decomposition of A with A = L.LT . Thus, for simulating

correlated asset prices, we must use Cholesky decomposition.

44

8. PRICING MULTI-ASSET AMERICAN OPTIONS

In this section, we price three exotic high-dimensional American options that

exist in the literature. One of them is a spread option and the other two are basket

options on two and five assets. For options on the maximum of k assets, the results

calculated via our LSM algorithm are compared with several methods including LSM

algorithm used in the relevant reference. For the spread options, we only compare our

price estimates with those of the trinomial tree method.

8.1. Simulation of American-style Multi-asset Options

It is obvious that pricing high-dimensional options is more complicated for Ameri-

can versions due to the early exercise opportunity. The pricing methods such as lattices

and finite difference methods, which work well for American options on a single asset,

require work that is exponential in the number of state variables (Broadie and Glasser-

man, 1997).

Monte Carlo simulation has the distinct advantage of having a convergence rate,

which is independent of state variables (Broadie and Glasserman, 1997). Besides, it is

easy to handle different types of models and payoff structures.

In simulation of multi-asset options, correlation between the assets is a fact that

must not be neglected. By Cholesky decomposition, it is simple to simulate prices

of assets that are correlated and it almost does not require any extra computational

effort.

Altogether, due to the pros of Monte Carlo simulation regarding its convergence

rate and exponential increase of required work in the other methods, Monte Carlo

simulation is especially appropriate for pricing multi-asset derivatives.

45

8.2. LSM Algorithm for Multi-Asset American Options

As mentioned in Section 4.3, the LSM algorithm can be modified for multi-

dimensional cases. Longstaff and Schwartz (2001) price an option on the maximum of

five assets by using the LSM algorithm. They have obtained a result within the tightest

confidence interval calculated by the stochastic mesh method of Broadie and Glasser-

man (1997). Also, the LSM algorithm needs one to two minutes while the computation

of the stochastic mesh method takes about 20 hours.

It is possible and easy to modify the algorithm for multi-dimensional cases. The

addition in the algorithm for multi-asset options is to generate prices of more than one

asset and to extend the basis functions as they include prices of all assets. Of course,

while generating the asset prices, the correlation should be considered. However, not

much work is required to modify the algorithm for the multi-dimensional case.

Let X0, X1, . . . , Xd denote a <k valued Markov chain describing all relevant fi-

nancial information where X0 is the fixed initial information, k is the number of assets

and d is the number of exercisable time points. The algorithm given in 4.3 is still valid

for the multi-dimensional case. The only difference is in the dimension of the Markov

chain. To summarize:

i. Simulate prices of assets for n independent paths of the Markov chain

{X0, X1, . . . , Xd} by using Cholesky decomposition

ii. At terminal nodes, set continuation values to the payoff function,

iii. Apply backward induction: for i = d− 1, . . . , 1,

• Fit regression by using the value of the option at that time steps and basis

functions to estimate the coefficients.

• Estimate the continuation values according to the estimated coefficients by

Equation 4.1.

• Compare the estimated continuation values and the income from early ex-

ercise, and decide to exercise or not

iv. Take the average of the incomes in the paths and discount it to find the price of

46

the option.

8.3. American Options on the Maximum of Two Assets

We gave brief information about American options on the maximum of k assets

in Section 7.1.1. This section contains the explanation of the possible control variable

and R implementation for pricing an American option on the maximum of two assets.

It also includes the numerical results on the example analyzed by Glasserman (2003).

8.3.1. Price of European Option on the Maximum of Two Assets as Control

Variable

The price of an American option is related to the price of its European equivalent.

Since it is possible to calculate its exact value, the price of a European option can

be used as control variable while applying CV to pricing an American option. The

equations from 7.1 to 7.7 given in Section 7.1.1.1 enable us to compute the exact price

of a European option on the maximum of two assets. Hence, for the American options

on the maximum of two assets, European style option can be selected as control variable

in simulation. Our R implementation includes the application of CV with this control

variable.

8.3.2. R Implementation

In the R implementation of pricing American options on the maximum of two

assets, beside the antithetic variates, we also used the control variates. Since we know

the closed form solution of the European equivalent of the option we price, it can be

used as control variable. Chapter 5 of Haug (2007) includes the formula of the price

of a European option on the maximum of two assets. The key in this simulation is the

determination of the parameter c. It is possible to calculate c according to Equation

3.11 by using the payoff of the American option and the payoff of the European option,

which are the results of a pilot study simulation with a smaller size of sample, for

example a simulation with 1000 paths. However, to shorten the R code, we define

47

parameter c as an input to the function instead of calculating it by a pilot study

within the function. Appendix A.8 includes the implementation that is explained in

this section. The version with the pilot study can be found in Appendix A.9.

The function simul US max of 2 prices an American call option on the maximum

of two assets by using LSM algorithm. In the R implementation, we first calculate the

time interval. Since we do not use polynomials, there is no need to normalize all val-

ues by dividing with the exercise price. Next, according to the given correlation, the

correlation matrix is formed. By using the Cholesky decomposition of the correlation

matrix, we convert the uncorrelated random variates into correlated ones. For this pur-

pose, we generate standard normal variates and store them in a matrix. The matrix

multiplication of L, which is transpose of Cholesky decomposition of correlation ma-

trix, and the matrix z, which consists of independent variates, gives us the correlated

variates. Notice that we use the function chol for Cholesky decomposition and syntax

%*% for the multiplication of matrices. Since we do not need the independent variates

anymore, the correlated ones are stored in z. After these operations, we generate the

terminal prices of assets by using the new matrix z. Different to the one-dimensional

case, we include the dividend terms of each asset while simulating the prices. In addi-

tion, the stock prices generated by antithetic variates are saved in the same vector for

easy calculations. Before the for loop, we assign the payoff to array CC, which will be

discounted for being an input for regression at time td−1.

Within the for loop, we first simulate asset prices at that time step. Therefore, we

again generate independent standard normal variates and transform them by using the

matrix L. After the simulation of asset prices in the current time step, in-the-money

points are determined to fit regression. Similar to the one-dimensional implementa-

tion, discountedCC, which is composed of discounted cash-flows, and basis functions

are formed to be input for the regression to be fitted. The continuation values are es-

timated according to the regression coefficients and the comparison between estimated

continuation values and income from the early exercise is made. CC is updated due to

the comparison. To go to the previous time step, asset prices at the current time step

are assigned as future prices.

48

In the end of the for loop, the payoff of all paths and the option price are com-

puted. According to the control variable, payoff is controlled by using the parameter c

and the exact value of the control variable. Within the code, the function EU max of 2

calculates the exact price of the European equivalent. Then, the price estimate found

by using the controlled payoff ”payoff.c” and the standard error of the simulation are

printed out.

Meanwhile, it should be mentioned that instead of using Cholesky decomposition,

a function named mvrnorm, which is defined in the R package MASS, could be used.

The function mvrnorm generates samples from the multivariate normal distribution;

inputs are the dimension, mean, standard deviation and the variance-covariance matrix.

Of course this function uses Cholesky decomposition as well.


We use Example 8.6.1 from Glasserman (2003), which is also used in Andersen

and Broadie (2004) and Broadie and Glasserman (2004). It is a call option on the

maximum of two assets, S1 and S2. Hence, its payoff function h is;

h(S1, S2

)= max

(max

(S1, S2

)−K, 0

)The initial asset prices are equal to each other and the exercise price K is three. The

risk-free interest rate r is 0.05. Volatilities and dividend yields, which are considered

as continuous, are same for each asset and are sequentially 0.2 and 0.1. It is assumed

that there is no correlation between the assets. This means they are independent of

each other. The option has nine equally spaced exercisable dates and expires in three

years.

Glasserman (2003) uses three cases by varying the starting asset prices. The

summarized results can be seen in Table 8.1. In Table 8.1, the column LSM includes

our price estimates by using seven basis functions; a constant, asset prices, squares

of asset prices, multiplication of asset prices and the payoff function. According to

49

Table 8.1. Price estimates for a two asset American maximum call option

Closed Form Binomial VRF VRF

S0 European Model 95% CI LSM (AV) (AV & CV)

90 6.5551 8.075 [8.053, 8.082] 8.0598 2.487066 4.15552

100 11.1957 13.902 [13.892, 13.934] 13.9001 2.747369 4.023047

110 16.9286 21.345 [21.316, 21.359] 21.320 3.109262 3.938483

Glasserman, this choice of basis functions is the set with the minimum number of

elements that gives the best results. Furthermore, closed form European prices are

computed by the formula in Haug (2007) and the 95 percent confidence interval (CI) is

the interval computed by Andersen and Broadie (2004). Besides these, price estimates

of the binomial tree are calculated by using the multi-asset binomial method of Boyle et.

al (1989) which has a error of 0.003. Moreover, VRF indicates the variance reduction

factor for applying antithetic variates (AV) and control variates (CV) to the LSM

algorithm. The closed form European price is used as control variable for each case,

assuming that it is the exact value. In the table, VRFs when only AV is applied and

when both AV and CV are applied are given separately.

According to Table 8.1, our price estimates with LSM algorithm, which have

standard error of 0.002, are in the 95 percent confidence interval computed by Andersen

and Broadie (2004). Also, as it can be noticed that AV is a more effective variance

reduction method when the option is deeply in the money due to the initial starting

asset prices. If the option starts out of the money, then simulation results in a lower

variance reduction factor. In addition, CV is not as efficient as AV. However, when it

is applied, it decreases the variance as well. In opposition to AV, the efficiency of CV

decreases when the option starts deeply in the money.

It is important to stress that in the example Glasserman (2003) priced, there

is no correlation between assets. However, our implementation given in Section 8.3.2

is capable of pricing any American option on the maximum of two assets even when

the assets are correlated. In addition, the variance reduction factor of the simulation

50

increases as assets are correlated.

8.3.3.1. Early Exercise Region. The early exercise region of an option written on more

than one asset is getting smaller as we go backwards in time. However, LSM algorithm

fits the regression by using all in-the-money points, which is the exercise region at

maturity.

Figure 8.1. Early exercise region of an American call option on the maximum of two

assets

51

Figure 8.1 shows the early exercise region of an American option on the maxi-

mum of two assets at different time points. It can be seen that the exercise region is

composed of two sub regions. Furthermore, as we get closer to the start, the exercise

region is getting smaller and the sub regions are getting farer apart from each other.

The inefficiency of the LSM algorithm related to the selection of input points to the

regression can also be seen from the plots in Figure 8.1. How to reduce the area used

for the regression is an open question.

8.3.3.2. Number of Time Steps versus Early Exercise Premium. It was mentioned ear-

lier that a Bermudan option, which has discrete early exercise opportunities, is placed

between American and European options. As we price Bermudan options by simula-

tion, it might be possible to determine the number of time steps necessary that the

price of the Bermudan option gets equal to the American price.

Since LSM algorithm estimates the option price low, the number of time steps

which results in the maximum early exercise premium might be selected for pricing.

So, Bermudan option we priced might be closer to its American equivalent. Also, if

we know the early exercise premium regarding to the early exercise property, it will be

easy to find the price of the American option by using the closed form solution of the

European option and the early exercise premium.

Figure 8.2 shows the early exercise premium of the case we priced in the previous

section for different numbers of time steps. The algorithm is run with 50000 antithetic

pairs and the estimates have standard errors ranging from 0.006 to 0.008. As it is

seen in Figure 8.2, the increase in early exercise premium is not proportional to the

increase in the number of time steps. As we consider more time steps in simulation,

the pace of increase in the premium decreases and the premium converges to a value.

The simulation time is linearly increases with the number of time steps. Hence, we do

not want to allow too many exercise opportunities in the algorithm due to the very

small changes in the premium. Broadie and Glasserman (2004) price this option by

considering nine time steps and call it as American. This experiment shows that there

52


American call option on the maximum of two assets

is a significant difference for the early exercise premium. Considering nine time steps

results in a premium of 2.667936 but as it is 2.838814 when the option is priced with

45 time steps. Notice that more than 45 time steps increases the computational cost

without changing the price.

It was mentioned that there is a closed form solution for the European option

on the maximum of two assets. As the price of the American option is the sum of

the price of its European equivalent and the early exercise premium, it is possible to

find the price of the American option by using the premium. Table 8.2 summarizes the

53

Table 8.2. Price estimates of binomial tree and LSM with different number of time

steps for a two asset American maximum call option on two assets

S0 Binomial Tree LSM LSM∗

90 8.075 8.0598 8.152799

100 13.902 13.9001 14.03451

110 21.345 21.32 21.53642

price estimates for the same American maximum call option, which was priced by using

nine time steps. The estimates in the column LSM are results calculated earlier (with

nine time steps) whereas the ones in the column LSM∗ are calculated with 45 time

steps. There is a significant differences in the estimated prices whatever the initial

asset price is. The values calculated by LSM∗ are higher. Since we know that the

American option price is larger than the Bermudan one, it shows that there is a better

convergence of Bermudan option to its American equivalent if we use 45 time steps. In

other words, LSM estimates more precisely when 45 time steps are used.

8.4. American Options on the Maximum of Five Assets

After pricing an American option on the maximum of two assets, we consider

the case where the payoff is on the maximum of five assets. Longstaff and Schwartz

(2001) compare their results with the results of Broadie and Glasserman (1997) when

they assert that their LSM algorithm works well for multi-dimensional cases. The

implementation of this type of option in R differs from the implementation of the option

on the maximum of two assets only in the number of assets and the basis functions

used. Because of its length, the implementation using the basis functions stated by

Longstaff and Schwartz (2001) with AV is given in Appendix A.10. Numerical results

are included in this section.

54


Broadie and Glasserman (2004) apply their stochastic mesh method to an Amer-

ican call option on the maximum of five assets, which has the payoff function:

h(S1, S2, S3, S4, S5

)= max

(max

(S1, S2, S3, S4, S5

)−K, 0

)It is assumed that the assets are not correlated. The option has a three year life and has

nine exercise opportunities, in other words, exercisable three times per year. Volatility

of each asset is 20 percent. The assets pay 10 percent continuous dividend. The risk

free rate is 5 percent and the strike price is 100. The initial stock prices are assumed

to be the same for all assets and taken as 90,100 and 110.

Table 8.3 shows the Monte Carlo simulation results for the price of the European

option, the confidence interval for the American option, which is computed by Andersen

and Broadie (2004), our price estimates for the American option, which are calculated

by LSM algorithm using 50000 paths and 19 basis functions as described in the paper

of Longstaff and Schwartz (2001). These 19 basis functions are a constant, the first five

Hermite polynomials in the maximum values of the five assets, the four asset prices and

squares of the prices of the second through fifth highest asset prices, the product of the

highest and second highest, second highest and third highest, etc. The price estimates

of LSM algorithm have standard errors ranging from 0.004 to 0.007. It is remarkable

to notice that LSM estimates are within the 95 percent confidence interval. This was

the example used by Longstaff and Schwartz (2001) to show that their algorithm works

well for multi-dimensional options. In addition, VRFs reached by using AV can be seen

in the table.

It should be remarked that since we do not know the exact price of the European

equivalent of the option, we cannot apply CV. In addition, like for the maximum option

on two assets, our implementation is also working when the asset prices are correlated.

55

Table 8.3. Price estimates for an American call option on the maximum of five assets

S0 Monte Carlo EU 95% CI LSM VRF

90 14.581 [16.602, 16.655] 16.6159 2.324062

100 23.0493 [26.109, 26.292] 26.1015 2.390550

110 32.6993 [36.704, 36.832] 36.725 2.39652

Table 8.4. Price estimates for an American call option on the maximum of five assets

with different number of time steps

S0 Monte Carlo EU 95% CI LSM LSM∗

90 14.581 [16.602, 16.655] 16.6159 16.89765

100 23.0493 [26.109, 26.292] 26.1015 26.42993

110 32.6993 [36.704, 36.832] 36.725 37.1323

8.4.1.1. Number of Time Steps versus Early Exercise Premium. We question how many

exercise opportunities should be considered in pricing American options on the maxi-

mum of five assets. Figure 8.3 shows the early exercise premium for different numbers

of time steps. According to Figure 8.3, it is clear that, like in maximum option on

two assets, as the number of time steps increases, the increase in the premium gets

slower and it almost converges to a fixed value when the number of time steps is 45.

However, Broadie and Glasserman (2004) and Longstaff and Schwartz (2001) price this

American option by using only nine time steps and there is a significant increase in

premium between pricing by considering nine time steps and 45 time steps.

For the price of the option to reflect the American property, we price it with 45

exercise opportunities. The results can be seen in Table 8.4, in the column named

LSM∗. The values in the column LSM are estimates with nine time steps. Since we

price American options in simulation as Bermudan, the results of LSM∗ are closer to

be American than the estimates of Broadie and Glasserman (2004) and Longstaf and

Schwartz (2001).

56


American call option on the maximum of five assets

8.5. American Spread Options

In Section 7.1.2, we described spread options and gave information about their

application area. This section includes the R implementation of the simulation of

American spread options and numerical results on a case priced by Haug (2007). We

analyze this case to find a set of basis functions that results in good price estimates

and to illustrate the early exercise region of an American spread option. In addition,

we make experiments to answer the question how many exercisable time steps should

be considered in pricing an American spread option to have a close to exact value

57

of the early exercise premium, which is the difference between the European and the

American price.

8.5.1. European Spread Option Price as Control Variable

There is no closed form solution for European spread option. However, Kirk

(1995) approximated the price of a European spread option. By Monte Carlo simula-

tion with a huge sample size, which is quite fast, it is possible to get a price estimate

for a European spread option that is close to the exact price. In this section, we com-

pare price estimates of binomial tree, Kirk’s approximation and simulation to discuss

whether one of them can be used as control variable in pricing American spread options.

The R implementation of pricing a European spread call by simulation is included in

Appendix A.11.

Table 8.5 summarizes the price estimates of binomial tree, which is coded by Haug

(2007), approximation of Kirk (1995) and Monte Carlo simulation with 10 replication

of 1000000 paths for a European spread option with initial stock prices S10 = 122 and

S20 = 120. The quantitaties of the stocks are equal to one. The life of the option, the

correlation between assets and the volatilities are varied within the table.

According to Table 8.5, price estimates of Kirk’s approximation and Monte Carlo

simulation are close to each other for all cases. We know that Monte Carlo simulation

converges to the true option value when a huge sample size is used. Thus, Kirk’s

approximation, which gives almost identical results with the simulation, can be assumed

as the exact price of the European spread option and it allows us to use the European

spread option price as the control variable in pricing an American spread option. Hence,

in the implementation of pricing American spread options with control variates, we will

use Kirk’s approximation as the exact price of the European spread option.

58

Table 8.5. Price estimates of European spread call using Binomial tree, Kirk’s

approximation and Monte Carlo simulation

T=0.5 T=1

Method σ1 σ2 ρ = −0.5 ρ = 0 ρ = 0.5 ρ = −0.5 ρ = 0 ρ = 0.5

Tree 10.7566 8.708 6.0286 14.6148 11.8706 8.2824

Kirk’s Approx. 0.20 0.20 10.7517 8.70198 6.02572 14.61008 11.86839 8.278906

Simulation 10.751 8.70067 6.02693 14.6068 11.86757 8.276589

Tree 12.2031 9.9377 7.0097 16.5453 13.521 9.5992

Kirk’s Approx. 0.25 0.20 12.1941 9.93397 7.00672 16.53485 13.51713 9.595823

Simulation 12.1927 9.93385 7.00648 16.53468 13.51764 9.592516

Tree 12.1521 9.8811 6.9323 16.4762 13.4451 9.4954

Kirk’s Approx. 0.20 0.25 12.1483 9.87802 6.92836 16.47387 13.4423 9.490681

Simulation 12.1482 9.87698 6.92588 16.47266 13.44079 9.490336

8.5.2. R Implementation

R codes to price American spread options without a pilot study for CV can be

found in Appendix A.12. The function simul US spread price an American spread

option by using the LSM algorithm. This implementation does not differ much from

the early implementations of pricing other types of multi-asset options. The antithetic

variates method cannot be used due to the payoff structure. However, the price of the

European spread option is used as control variable to apply CV and its exact value is

assumed to be the result of Kirk’s approximation. The function takes the parameter c,

for CV, as an input. The version with the pilot study can be seen in Appendix A.13.

We first calculate the time interval before forming the correlation matrix. Using

Cholesky decomposition of the correlation matrix, the uncorrelated random variates

are converted into correlated. After the generation of the asset prices, the payoff is

assigned to the array CC, which will be an input for regression at time td−1 after the

discounting. Before the for loop, the payoff of the European style is stored to use it

while applying CV.

59

The content of the for loop within the implementation is, except for the payoff

structure, similar to the loops in the other implementations. It includes the generation

of the asset prices by using the Cholesky decomposition, fitting the regression according

to the basis functions, the comparison between the estimated continuation values and

the immediate income from payoff and the assignment of the asset prices for the next

step of the loop.

In the end of the loop, the payoff is calculated using the parameter c and the

exact value of the options European equivalent. The exact value is accepted to be the

result of Kirk’s approximation. The function kirk approximation computes the price

of the European spread option. After CV calculations, the function prints the price

estimate and the standard error of the simulation.


Haug (2007) price an American spread call option, which has payoff structure

as in equation 7.8, by using trinomial tree with 100 time steps. The option is on two

assets which have initial prices S10 = 122, S2

0 = 120 with quantities Q1 = Q2 = 1 and

the exercise price K is three. The risk free interest rate r is 0.1 and the cost of carry

of both assets is zero. It is essential to notice that the cost-of-carry is the difference

between the risk-free interest rate r and the continuous dividend yield δ. Thus, if the

cost-of-carry is zero, it means dividend yield is equal to the risk-free interest rate. The

option is priced with 50 exercise opportunities by varying the time to expiry as well as

the correlation between assets and their volatilities.

First, we make experiments for the selection of the basis functions. After illus-

trating the exercise region, we investigate how many exercisable time points should be

considered.

8.5.3.1. Basis Functions. In the experiments, we looked for the best set of basis func-

tions that ends with a price estimate close to the result of the trinomial tree in the

60

book of Haug (2007). The sets of basis functions tried in the experiments are varied

by the payoff function, prices of the assets, their squares and cross-products. These

sets can be seen in Table 8.6. The sets that include the payoff as a basis function do

not contain the price of the second asset because S1, S2 and h (S1, S2) are linearly

dependent.

Table 8.6. Sets of basis functions used in pricing an American spread call

Sets Basis Functions

1 h(S1, S2), S1, S1S2

2 h(S1, S2), S1, (S1)2, S1S2

3 S1, S2, S1S2

4 S1, S2, (S1)2, (S2)2, S1S2

5 S1, S2, (S1)2, (S2)2, S1S2, (S1)2S2, S1(S2)2

6 S1, S2, (S1)2, (S2)2, S1S2, (S1)2S2, S1(S2)2, (S1)2(S2)2

The price estimates for the American spread call option achieved by using different

sets of basis of functions are given in Table 8.7. It is seen that the sixth set, which is

composed of the asset prices, squares of the asset prices and three cross-multiplication

of asset prices is the appropriate choice for basis functions because we know that LSM

algorithm estimate the price low and this choice of basis functions results in better

price estimates. Also, the estimates of the fifth set are close to the those of the sixth.

Thus, to reduce the computational time even more, the fifth set may be chosen for

valuation, however, the decrease in time consumed by using less basis functions is not

significant. As a result of this experiment, we will continue pricing American spread

options by using the basis functions of the sixth set. Furthermore, the price estimates

of LSM with the sixth set are not identical to those of trinomial tree. However, we

do not know whether the results of the trinomial tree are 100 percent exact values of

the option. Also, it estimates the price of European option always higher than Kirk’s

approximation and simulation as shown in Section 8.5.1.

8.5.3.2. Variance Reduction. After explanaining why it is possible to use a European

spread option as control variable under the assumption its exact price is the result

61

Table 8.7. Price estimates for American spread call option by using different sets of

basis functions in pricing algorithm

T=0.5 T=1

Method σ1 σ2 ρ = −0.5 ρ = 0 ρ = 0.5 ρ = −0.5 ρ = 0 ρ = 0.5

LSM - Set1 10.83578 8.755928 6.064245 14.88673 12.10921 8.422175

LSM - Set2 10.8104 8.745786 6.056096 14.90784 12.08815 8.449647

LSM - Set3 0.20 0.20 10.83052 8.756744 6.065066 14.8974 12.10153 8.424005

LSM - Set4 10.85715 8.780862 6.079356 14.96019 12.13907 8.468816

LSM - Set5 10.87056 8.7908 6.08445 14.98109 12.16715 8.487024

LSM - Set6 10.87228 8.798006 6.085901 14.98986 12.170113 8.48866

Tree 10.8754 8.8029 6.0939 14.9967 12.178 8.4952

LSM - Set1 12.28299 10.01362 7.05683 16.91463 13.78644 9.77719

LSM - Set2 12.26572 10.00309 7.044334 16.8846 13.77136 9.774053

LSM - Set3 0.25 0.20 12.27982 10.014409 7.054606 16.91015 13.78192 9.77025

LSM - Set4 12.31974 10.03117 7.072127 16.93783 13.84705 9.82853

LSM - Set5 12.33751 10.04854 7.078977 16.97063 13.85917 9.839035

LSM - Set6 12.33544 10.0432 7.07943 16.97401 13.86069 9.840375

Tree 12.3383 10.0468 7.0858 16.9789 13.8727 9.8463

LSM - Set1 12.24264 9.944552 6.972998 16.81626 13.69899 9.671492

LSM - Set2 12.22076 9.903828 6.967923 16.78713 13.68161 9.656747

LSM - Set3 0.20 0.25 12.24853 9.940646 6.97869 16.82085 13.69681 9.668236

LSM - Set4 12.26254 9.97314 6.98868 16.87373 13.7772 9.711424

LSM - Set5 12.28186 9.981732 6.99179 16.90122 13.77125 9.72888

LSM - Set6 12.28264 9.98341 6.996226 16.90453 13.77939 9.725018

Tree 12.2867 9.9897 7.0082 16.9085 13.7953 9.7415

of Kirk’s approximation, we report the variance reduction factors (VRFs) that are

achieved by this method. It should be remarked that antithetic variates are not used

for spread options as they do not lead to any variance reduction due to the structure

of the payoff function.

Table 8.8 includes the VRFs for the American spread option we price. VRFs

change according to the life time of the option, the correlation between the underlying

assets and the volatilities. The data in Table 8.8 shows that VRF is negatively corre-

62

Table 8.8. VRFs in pricing American spread options

T=0.5 T=1

σ1 σ2 ρ = −0.5 ρ = 0 ρ = 0.5 ρ = −0.5 ρ = 0 ρ = 0.5

0.20 0.20 4.125466 3.925507 3.894836 3.416155 3.398603 3.340055

0.25 0.20 4.217281 3.987225 3.963501 3.48033 3.437527 3.4113

0.20 0.25 3.994254 3.759955 3.712589 3.2358 3.214024 3.165177

lated with the life of the option. As the life of the option increases, the reduction in

variance decreases. It is also clear that in all cases, the correlation between assets has

a decreasing influence on VRF. It means that if the correlation between the assets is

greater than zero, the simulation ends in a VRF that is relatively smaller in the case

where the assets are not correlated. On the contrary, the reduction in variance tends

to increase if the assets are correlated negatively. Furthermore, the results show that

the volatilities of the assets affect the VRF. When the volatility of the first asset, which

has the positive sign in the payoff function of a call, is smaller than the volatility of

the second asset, the simulation results in the lowest VRFs. If it is assumed that the

volatility of the first asset is greater, then we get the highest VRFs.

8.5.3.3. Early Exercise Region. After the implementation of the algorithm in R, it is

not hard to have a plot of asset prices where the early exercise takes place, in other

words, a plot of the exercise region. Figure 8.4 shows exercise region of an American

spread call option at time td−1. Parameters of the option are assumed to be same as

in the example Haug (2007) considered. The line in Figure 8.4 represents the exercise

boundary at the expiration date. Since the payoff is as in Equation 7.8, the condition

for the option to be exercised at time T is

S1 > S2 +K

which makes the payoff positive. Taking this fact into account we can conclude that

the exercise region is getting farer away from the line, which shows that the price of

the first asset is equal to the sum of the price of the second asset and the exercise price,

63

as the time to expiration increases. In other words, the exercise region is moving to

the right hand side. This means the difference between the prices of the assets should

be large to exercise early.

Figure 8.4. Early exercise region of an American spread call option at time td−1

As we know, LSM algorithm takes all in-the-money points as input for the re-

gression. This is the area to the right hand side of the straight line (for calls) in Figure

8.4. So, LSM algorithm uses all the paths that are to the right hand side of the line.

However, it is obvious that for early exercise, the payoff should be larger as we go

backwards in time. Thus, to consider all paths in regression leads to computational

cost. It might be possible to fit regression in a smaller area to specify the exercise

boundary. For calls, at a time step, the minimum difference between the prices of the

64

assets that early exercise takes place can be used as the exercise boundary at that time

step. How to reduce the region that is considered as input for regression to determine

this boundary is an open question.

8.5.3.4. Number of Time Steps versus Early Exercise Premium. Similarly to the idea

for American options on the maximum of two assets, we should consider the number

of time steps in the pricing algorithm necessary to obtain price estimates that are

sufficiently close to the true value of an American spread option.

Figure 8.5. Plots of early exercise premium versus number of time steps for American

spread call options

Figure 8.5 contains two plots which show early exercise premium for different

number of time steps that are powers of two as well as the premium with 50 time steps

per year. The left plot is for the life of the option equal to 0.5 years whereas it is

one year for the right plot. In both cases, volatilities of the assets are 20 percent and

there is no correlation between the asset prices. According to Figure 8.5, it is seen

that increasing the total number of time steps considered in pricing algorithm does not

always improve the price estimate and that the early exercise premium converges to

a certain value. Thus, considering more time steps than a specific number does not

65

significantly improve the price estimate for the American spread option. However, it

increases the computational cost. As it can be noticed from the figure, the sufficient

number of exercise opportunities per year is 50 under these conditions.

66

9. CONCLUSIONS

FD and tree techniques are efficient methods to price American options with sin-

gle underlying security whereas simulation works well better for multi-asset American

options. In this study, we focused on the LSM algorithm of Longstaff and Schwartz

(2001), which is a regression-based Monte Carlo simulation method for pricing Ameri-

can options.

When we know the early exercise boundary, pricing an American option is quite

easy. In the one-dimensional case, the estimate of LSM for the early exercise boundary

at the last time step does not match with the result of Black-Scholes formula and

Newton Raphson method. This shows that LSM cannot estimate the boundary well.

We tried to reduce the inefficiency of LSM algorithm about the input selection for the

regression and improved the algorithm. It now estimates the price of an American

option with less computational cost and more accurately than the LSM algorithm.

Furthermore, we coded an optimization approach, which maximizes the total value at

each time step, and we always had results higher than those of FD method. Therefore,

it might be possible to use it as an upper bound of the American option price.

For multi-asset American options, we extended the implementation of the LSM

algorithm. We analyzed how many early exercise points should be considered for the

option to be close to the American price and we showed that the price estimates in

the literature are actually too low for American options on the maximum of two or

five assets as the number of time steps was selected too small. For American spread

options, we tested different sets of basis functions and determined a useful one. We

also applied variance reduction techniques for multi-asset options. We showed that it is

possible to use Kirk’s approximation for European spread option as a control variable

on its American equivalent while applying control variates (CV). Considerable variance

reduction is achieved by using antithetic variates (AV) and CV for maximum options

on two assets, AV for maximum options on five assets and CV for spreads.

67

APPENDIX A: R CODES

A.1. R Codes for Black-Scholes Formula to Price European Options

EU black s c h o l e s<−function ( s0 ,K, r , sigma ,T, f l a g=”c” ){

# s0 . . . i n i t i a l a s s e t p r i c e

# K . . . s t r i k e Price

# r . . . r i s k−f r e e i n t e r e s t ra t e

# sigma . . . v o l a t i l i t y

# T . . . time hor i zon

# f l a g . . . c f o r c a l l op t i ons

d1<−( log ( s0/K)+( r+sigma ˆ2/2)∗T)/ ( sigma∗sqrt (T) ) ;

d2<−d1−sigma∗sqrt (T) ;

i f ( f l a g==”c” ){

s0∗pnorm( d1)−K∗exp(−r∗T)∗pnorm( d2 )

} else {

K∗exp(−r∗T)∗pnorm(−d2)−s0∗pnorm(−d1 )

}

}

A.2. R Codes for the Implementation of the LSM Algorithm

s imul US LSM<− function (n , d , s0 ,K, sigma , r ,T){

s0<−s0/K

dt <− T/d

z<−rnorm(n)

s . t<−s0∗exp ( ( r − 1/2∗sigma ˆ2)∗T+sigma∗z∗ (Tˆ 0 . 5 ) )

s . t [ ( n+1):(2∗n ) ]<−s0∗exp ( ( r − 1/2∗sigma ˆ2)∗T−sigma∗z∗ (Tˆ 0 . 5 ) )

CC<−pmax(1− s . t , 0 )

68

payo f f eu<−exp(−r∗T)∗ (CC[ 1 : n]+CC[ ( n+1):(2∗n ) ] ) /2∗K

eupr i c e<−mean( payo f f eu )

for ( k in (d−1) :1){

z<−rnorm(n)

mean<−( log ( s0)+k∗log ( s . t [ 1 : n ] ) ) / ( k+1)

vo l<−( k∗dt/ ( k +1))ˆ0.5∗z

s . t 1<−exp(mean+sigma∗vo l )

mean<−( log ( s0)+k∗log ( s . t [ ( n+1):(2∗n ) ] ) ) / ( k+1)

s . t 1 [ ( n+1):(2∗n ) ]<−exp(mean−sigma∗vo l )

CE<−pmax(1− s . t 1 ,0)

idx<−( 1 : ( 2 ∗n ) ) [CE>0]

discountedCC<− CC[ idx ] ∗exp(−r∗dt )

b a s i s 1<−exp(−s . t 1 [ idx ] /2)

b a s i s 2<−b a s i s 1∗(1− s . t 1 [ idx ] )

b a s i s 3<−b a s i s 1∗(1−2∗s . t 1 [ idx ]+( s . t 1 [ idx ] ˆ 2 )/2)

p<−glm( discountedCC˜b a s i s 1+b a s i s 2+b a s i s 3 )$coef f ic ients

estimatedCC<−p [1 ]+p [ 2 ] ∗b a s i s 1+p [ 3 ] ∗b a s i s 2+p [ 4 ] ∗b a s i s 3

EF<−rep (0 ,2∗n)

EF[ idx ]<−(CE[ idx ]>estimatedCC )

CC<−(EF==0)∗CC∗exp(−r∗dt)+(EF==1)∗CE

s . t<−s . t 1

}

payo f f<−exp(−r∗dt )∗ (CC[ 1 : n]+CC[ ( n+1):(2∗n ) ] ) /2

u s p r i c e<−mean( payo f f∗K)

e r r o r<−1 .96∗sd ( payo f f∗K)/sqrt (n)

ea r l y ex<−uspr i ce−eupr i c e

data . frame ( uspr i ce , e r ro r , eupr ice , premium )

}

69

A.3. R Codes for the Exact Early Exercise Boundary of an American Put

at time td−1 by Newton-Raphson Method

newton raphson<−function ( i n i t i a l . s ,K, r , sigma , dt ){

# i n i t i a l . . . i n i t i a l va lue f o r the a l gor i thm

# K . . . s t r i k e Price

# r . . . r i s k−f r e e i n t e r e s t ra t e

# sigma . . . v o l a t i l i t y

# dt . . . l e n g t h o f time i n t e r v a l

s<− i n i t i a l . s

repeat{

d1=(log ( s/K)+( r+sigma ˆ2/2)∗dt )/ ( sigma∗sqrt (dt ) ) ;

d2=d1−sigma∗sqrt (dt )

f<−(K∗exp(−r∗dt )∗pnorm(−d2)−s∗pnorm(−d1)−(K−s ) )

f d e r i v a t i v e<−( ( s∗sigma∗sqrt (dt ))ˆ(−1))

∗(−K∗exp(−r∗dt )∗dnorm( d2)+s∗dnorm( d1))+pnorm( d1 )

new . s<−s−f/ f d e r i v a t i v e

conv<−abs (new . s−s )

i f ( ( conv/abs ( s ))<1e−8) break

s<−new . s

}

s

}

A.4. R Codes for the Implementation of the SRA

s imul US SRA<−function (n , np i l o t , nreg , d , s0 ,K, sigma , r ,T){

s0<−s0/K

dt <− T/d

z<−rnorm(n)

70

s . t<−s0∗exp ( ( r − 1/2∗sigma ˆ2)∗T+sigma∗rnorm(n)∗ (Tˆ 0 . 5 ) )


CC<−pmax(1− s . t , 0 )

pthresho ld<−NULL

pthresho ld [ d ]<−1

payo f f eu<−exp(−r∗ (d)∗dt )∗ (CC[ 1 : n]+CC[ ( n+1):(2∗n ) ] ) /2∗K


for ( k in (d−1) :1){

z<−rnorm(n)


vo l<−( k∗dt/ ( k +1))ˆ0.5∗z

s . t 1<−exp(mean+sigma∗vo l )


s . t 1 [ ( n+1):(2∗n ) ]<−exp(mean−sigma∗vo l )

CE<−pmax(1− s . t 1 ,0)

###TO HAVE AN IDEA ABOUT THRESHOLD###

idx<−( 1 : ( 2 ∗n ) ) [CE>0]

idx<−idx [ 1 : n p i l o t ]

discountedCC<− CC[ idx ] ∗exp(−r∗dt )

b a s i s 1<−exp(−s . t 1 [ idx ] /2)

b a s i s 2<−b a s i s 1∗(1− s . t 1 [ idx ] )

b a s i s 3<−b a s i s 1∗(1−2∗s . t 1 [ idx ]+( s . t 1 [ idx ] ˆ 2 )/2)

p<−glm( discountedCC˜b a s i s 1+b a s i s 2+b a s i s 3 )$coef f ic ients


p i l o t .EF<−(CE[ idx ]>estimatedCC )

pthresho ld [ k ]<−max( s . t 1 [ idx ] [ p i l o t .EF==1])

i f ( a l l ( p i l o t .EF==0)){

pthresho ld [ k ]<−pthresho ld [ k+1]

}

####REGRESSION AROUND THAT THRESHOLD VALUE###

distance<−abs ( s . t 1−pthresho ld [ k ] )

e p s i l o n<−sort ( distance ) [ nreg ]

71

idx<−( 1 : ( 2 ∗n ) ) [ s . t 1>=(pthresho ld [ k]− e p s i l o n )&

s . t 1<=(pthresho ld [ k]+ e p s i l o n ) ]

discountedCC<−CC[ idx ] ∗exp(−r∗dt )

p<−lm( discountedCC˜s . t 1 [ idx ] ) $coef f ic ients

estimatedCC<−p [1 ]+p [ 2 ] ∗s . t 1 [ idx ]

EF<−rep (0 ,2∗n)

EF[ s . t 1<( pthre sho ld [ k]− e p s i l o n ) ]<−1



s . t<−s . t 1

}

payo f fu s<−exp(−r∗dt )∗ (CC[ 1 : n]+CC[ ( n+1):(2∗n ) ] ) /2∗K

u s p r i c e <− mean( payo f fu s )

e r r o r <− 1 .96∗sd ( payo f fu s )/sqrt (n)

data . frame ( uspr i ce , e r ro r , eupr i c e )

}

A.5. R Codes for the Implementation of the Optimization Approach

s imul US OPT<−function (n , d , s0 ,K, sigma , r ,T){

dt <− T/d

z<−rnorm(n)

s . t<−s0∗exp ( ( r − 1/2∗sigma ˆ2)∗T+sigma∗z∗ (Tˆ 0 . 5 ) )


CC<−pmax(K−s . t , 0 )

opt imized<−NULL

optimized [ d ]<−1

for ( k in (d−1) :1){

z<−rnorm(n)


vo l<−( k∗dt/ ( k +1))ˆ0.5∗z∗sigma

72

s . t 1<−exp(mean+vol )


s . t 1 [ ( n+1):(2∗n ) ]<−exp(mean−vo l )

CE<−pmax(K−s . t 1 ,0)

o<−order ( s . t 1)

cont inuat i on<−CC[ o ] ∗exp(−r∗dt )

e x e r c i s e<−CE[ o ]

t o t a l v a l u e<−cumsum( cont inuat i on [ ( 2 ∗n ) : 1 ] ) [ ( 2 ∗n ) : 1 ]

+cumsum( e x e r c i s e )− cont inuat i on

opt imized [ k ]<−s . t 1 [ o ] [ which .max( t o t a l v a l u e ) ]

EF<−rep (0 ,2∗n)

EF[ s . t 1<=optimized [ k ] ]<−1


s . t<−s . t 1

}

payo f f<−exp(−r∗dt )∗ (CC[ 1 : n]+CC[ ( n+1):(2∗n ) ] ) /2

data . frame (mean( payo f f ) , sd ( payo f f )/sqrt (n ) )

}

A.6. R Codes for the Exact Price of a European Call on the Maximum of

Two Assets

EU max o f 2<−function ( s1 , s2 ,K, sigma1 , sigma2 , b1 , b2 , r ,T, rho ){

# This func t i on r e qu i r e s the package ”mnormt”

sigma<−sqrt ( sigma1ˆ2+sigma2ˆ2−2∗rho∗sigma1∗sigma2 )

d<−( log ( s1/s2 )+(b1−b2+sigma ˆ2/2)∗T)/ ( sigma∗sqrt (T) )

y1<−( log ( s1/K)+(b1+sigma1 ˆ2/2)∗T)/ ( sigma1∗sqrt (T) )

y2<−( log ( s2/K)+(b2+sigma2 ˆ2/2)∗T)/ ( sigma2∗sqrt (T) )

rho1<−( sigma1−rho∗sigma2 )/sigma

rho2<−( sigma2−rho∗sigma1 )/sigma

varcov<−matrix (c (1 , rep ( rho , 2 ) , 1 ) ,nrow=2)

73

varcov1<−matrix (c (1 , rep ( rho1 , 2 ) , 1 ) ,nrow=2)

varcov2<−matrix (c (1 , rep ( rho2 , 2 ) , 1 ) ,nrow=2)

s1∗exp ( ( b1−r )∗T)∗pmnorm(c ( y1 , d ) , rep ( 0 , 2 ) , varcov1 )

+s2∗exp ( ( b2−r )∗T)∗pmnorm(c ( y2 ,−d+sigma∗sqrt (T) ) , rep ( 0 , 2 ) , varcov2 )

−K∗exp(−r∗T)∗(1−pmnorm(c(−y1+sigma1∗sqrt (T) ,

−y2+sigma2∗sqrt (T) ) ,

rep ( 0 , 2 ) , varcov ) )

}

A.7. R Codes of the Kirk’s approximation for European Spread Options

k i rk approximation<−function ( S1 , S2 ,Q1=1,Q2=1,K,T, b1 , b2 , r ,

sigma1 , sigma2 , rho , f l a g=” c a l l ” ){

S<−Q1∗S1∗exp ( ( b1−r )∗T)/ (Q2∗S2∗exp ( ( b2−r )∗T)+K∗exp(−r∗T) )

F<−Q2∗S2∗exp ( ( b2−r )∗T)/ (Q2∗S2∗exp ( ( b2−r )∗T)+K∗exp(−r∗T) )

sigma<−sqrt ( sigma1ˆ2+(sigma2∗F)ˆ2−2∗rho∗sigma1∗sigma2∗F)

d1<−( log (S)+( sigma ˆ2/2)∗T)/ ( sigma∗sqrt (T) )

d2<−d1−sigma∗sqrt (T)

i f ( f l a g==” c a l l ” ){

(Q2∗S2∗exp ( ( b2−r )∗T)+K∗exp(−r∗T) ) ∗ (S∗pnorm( d1)−pnorm( d2 ) )

} else {

(Q2∗S2∗exp ( ( b2−r )∗T)+K∗exp(−r∗T) ) ∗ (pnorm(−d2)−S∗pnorm(−d1 ) )

}

}


without Pilot Study for CV

74

s imul US max o f 2<−function (n , d , s0 ,K, sigma1 , sigma2 ,

r , dividend ,T, rho , c ){

dt<−T/d

cormat<−matrix (c (1 , rep ( rho , 2 ) , 1 ) ,nrow=2)

L<−t ( chol ( cormat ) )

z<−matrix (rnorm(n∗2) ,nrow=2)

z<−L%∗%z

s1 . t<−s0∗exp ( ( r−dividend−1/2∗sigma1 ˆ2)∗T+sigma1∗ (Tˆ 0 . 5 )∗z [ 1 , ] )

s1 . t [ ( n+1):(2∗n ) ]<−s0∗exp ( ( r−dividend−1/2∗sigma1 ˆ2)∗T

−sigma1∗ (Tˆ 0 . 5 )∗z [ 1 , ] )



−sigma2∗ (Tˆ 0 . 5 )∗z [ 2 , ] )

CC<−pmax(pmax( s1 . t , s2 . t)−K, 0 )

payo f f eu<−exp(−r∗T)∗ (CC[ 1 : n]+CC[ ( n+1):(2∗n ) ] ) /2

for ( k in (d−1) :1){

z<−L%∗%matrix (rnorm(n∗2) ,nrow=2)

mean<−( log ( s0)+k∗log ( s1 . t [ 1 : n ] ) ) / ( k+1)

vo l<−( k∗dt/ ( k +1))ˆ0.5∗z [ 1 , ] ∗sigma1

s1 . t 1<−exp(mean+vol )

mean<−( log ( s0)+k∗log ( s1 . t [ ( n+1):(2∗n ) ] ) ) / ( k+1)

s1 . t 1 [ ( n+1):(2∗n ) ]<−exp(mean−vo l )


vo l<−( k∗dt/ ( k +1))ˆ0.5∗z [ 2 , ] ∗sigma1



s2 . t 1 [ ( n+1):(2∗n ) ]<−exp(mean−vo l )

CE<− pmax(pmax( s1 . t 1 , s2 . t 1)−K, 0 )

idx<−( 1 : ( 2 ∗n ) ) [CE>0]


b a s i s 1<−CE[ idx ]

75

b a s i s 2<−s1 . t 1 [ idx ]

b a s i s 3<−s1 . t 1 [ idx ]ˆ2

b a s i s 4<−s2 . t 1 [ idx ]

b a s i s 5<−s2 . t 1 [ idx ]ˆ2

b a s i s 6<−s1 . t 1 [ idx ] ∗s2 . t 1 [ idx ]

p<−glm( discountedCC˜b a s i s 1+b a s i s 2+b a s i s 3+b a s i s 4

+b a s i s 5+b a s i s 6 )$coef f ic ients


+p [ 5 ] ∗b a s i s 4+p [ 6 ] ∗b a s i s 5+p [ 7 ] ∗b a s i s 6

EF<−rep (0 ,2∗n)



s1 . t<−s1 . t 1 ; s2 . t<−s2 . t 1

}

payo f f<− exp(−r∗dt )∗ (CC[ 1 : n]+CC[ ( n+1):(2∗n ) ] ) /2

payo f f . c<−payof f−c∗ ( payof feu−EU max o f 2( s0 , s0 ,K, sigma1 , sigma2 ,

r−dividend , r−dividend , r ,T, rho ) )

u s p r i c e . c<−mean( payo f f . c )

s . e . c<−sd ( payo f f . c )/sqrt (n)

data . frame ( u s p r i c e . c , s . e . c )

}


with Pilot Study for CV

s imul US max o f 2 p i l o t<−function (n , d , s0 ,K, sigma1 , sigma2 ,

r , dividend ,T, rho ){

n p i l o t<−1000

dt<−T/d



76

### PILOT STUDY ###

z<−matrix (rnorm( n p i l o t ∗2) ,nrow=2)

z<−L%∗%z


s1 . t [ ( n p i l o t +1):(2∗ n p i l o t ) ]<−s0∗exp ( ( r−dividend−1/2∗sigma1 ˆ2)∗T

−sigma1∗ (Tˆ 0 . 5 )∗z [ 1 , ] )


s2 . t [ ( n p i l o t +1):(2∗ n p i l o t ) ]<−s0∗exp ( ( r−dividend−1/2∗sigma2 ˆ2)∗T

−sigma2∗ (Tˆ 0 . 5 )∗z [ 2 , ] )


payo f f eu<−exp(−r∗T)∗ (CC[ 1 : n p i l o t ]+CC[ ( n p i l o t +1):(2∗ n p i l o t ) ] ) /2

for ( k in (d−1) :1){

z<−L%∗%matrix (rnorm( n p i l o t ∗2) ,nrow=2)

mean<−( log ( s0)+k∗log ( s1 . t [ 1 : n p i l o t ] ) ) / ( k+1)

vo l<−( k∗dt/ ( k +1))ˆ0.5∗z [ 1 , ] ∗sigma1


mean<−( log ( s0)+k∗log ( s1 . t [ ( n p i l o t +1):(2∗ n p i l o t ) ] ) ) / ( k+1)

s1 . t 1 [ ( n p i l o t +1):(2∗ n p i l o t ) ]<−exp(mean−vo l )

mean<−( log ( s0)+k∗log ( s2 . t [ 1 : n p i l o t ] ) ) / ( k+1)

vo l<−( k∗dt/ ( k +1))ˆ0.5∗z [ 2 , ] ∗sigma1


mean<−( log ( s0)+k∗log ( s2 . t [ ( n p i l o t +1):(2∗ n p i l o t ) ] ) ) / ( k+1)

s2 . t 1 [ ( n p i l o t +1):(2∗ n p i l o t ) ]<−exp(mean−vo l )


idx<−( 1 : ( 2 ∗ n p i l o t ) ) [CE>0]



b a s i s 2<−s1 . t 1 [ idx ]

b a s i s 3<−s1 . t 1 [ idx ]ˆ2

b a s i s 4<−s2 . t 1 [ idx ]

77

b a s i s 5<−s2 . t 1 [ idx ]ˆ2




estimatedCC<−p [1 ]+p [ 2 ] ∗b a s i s 1+p [ 3 ] ∗b a s i s 2++p [ 4 ] ∗b a s i s 3


EF<−rep (0 ,2∗ n p i l o t )



s1 . t<−s1 . t 1 ; s2 . t<−s2 . t 1

}

payo f f<− exp(−r∗dt )∗ (CC[ 1 : n p i l o t ]+CC[ ( n p i l o t +1):(2∗ n p i l o t ) ] ) /2

c<−cov ( payof feu , payo f f )/var ( payo f f eu )

### PRICING WITH CV ###


z<−L%∗%z



−sigma1∗ (Tˆ 0 . 5 )∗z [ 1 , ] )



−sigma2∗ (Tˆ 0 . 5 )∗z [ 2 , ] )


payo f f eu<−exp(−r∗T)∗ (CC[ 1 : n]+CC[ ( n+1):(2∗n ) ] ) /2

for ( k in (d−1) :1){



vo l<−( k∗dt/ ( k +1))ˆ0.5∗z [ 1 , ] ∗sigma1



s1 . t 1 [ ( n+1):(2∗n ) ]<−exp(mean−vo l )

78


vo l<−( k∗dt/ ( k +1))ˆ0.5∗z [ 2 , ] ∗sigma1



s2 . t 1 [ ( n+1):(2∗n ) ]<−exp(mean−vo l )


idx<−( 1 : ( 2 ∗n ) ) [CE>0]



b a s i s 2<−s1 . t 1 [ idx ]

b a s i s 3<−s1 . t 1 [ idx ]ˆ2

b a s i s 4<−s2 . t 1 [ idx ]

b a s i s 5<−s2 . t 1 [ idx ]ˆ2






EF<−rep (0 ,2∗n)



s1 . t<−s1 . t 1 ; s2 . t<−s2 . t 1

}

payo f f<− exp(−r∗dt )∗ (CC[ 1 : n]+CC[ ( n+1):(2∗n ) ] ) /2

payo f f . c<−payof f−c∗ ( payof feu−EU max o f 2( s0 , s0 ,K, sigma1 , sigma2 ,

r−dividend , r−dividend , r ,T, rho ) )

u s p r i c e . c<−mean( payo f f . c )

s . e . c<−sd ( payo f f . c )/sqrt (n)

data . frame ( u s p r i c e . c , s . e . c )

}

79

A.10. R Codes to Price an American Call on the Maximum of Five Assets

s imul US max o f 5<−function ( s0 ,K, n , d , t , sigma , r , dividend , rho ){

s0<−s0/K

dt=t/d

cor matrix<−matrix (0 , ncol=5,nrow=5)

diag ( cor matrix )<−1

L<−t ( chol ( cor matrix ) )


z<−L%∗%z

s1 . t<−s0∗exp ( ( r−dividend−1/2∗sigma ˆ2)∗T+sigma∗ (Tˆ 0 . 5 )∗z [ 1 , ] )





s1 . t [ ( n+1):(2∗n ) ]<−s0∗exp ( ( r−dividend−1/2∗sigma ˆ2)∗T

−sigma∗ (Tˆ 0 . 5 )∗z [ 1 , ] )


−sigma∗ (Tˆ 0 . 5 )∗z [ 2 , ] )


−sigma∗ (Tˆ 0 . 5 )∗z [ 3 , ] )


−sigma∗ (Tˆ 0 . 5 )∗z [ 4 , ] )


−sigma∗ (Tˆ 0 . 5 )∗z [ 5 , ] )

CC<−pmax(pmax( s1 . t , s2 . t , s3 . t , s4 . t , s5 . t )−1 ,0)

for ( k in (d−1) :1){


s1 . t 1<−exp ( ( log ( s0)+k∗log ( s1 . t ) )/ ( k+1)

+sigma∗ ( k∗dt/ ( k +1))ˆ0.5∗z [ 1 , ] )


+sigma∗ ( k∗dt/ ( k +1))ˆ0.5∗z [ 2 , ] )

80


+sigma∗ ( k∗dt/ ( k +1))ˆ0.5∗z [ 3 , ] )


+sigma∗ ( k∗dt/ ( k +1))ˆ0.5∗z [ 4 , ] )


+sigma∗ ( k∗dt/ ( k +1))ˆ0.5∗z [ 5 , ] )

s1 . t 1 [ ( n+1):(2∗n ) ]<−exp ( ( log ( s0)+k∗log ( s1 . t [ ( n+1):(2∗n ) ] ) )

/ ( k+1)−sigma∗ ( k∗dt/ ( k +1))ˆ0.5∗z [ 1 , ] )


/ ( k+1)−sigma∗ ( k∗dt/ ( k +1))ˆ0.5∗z [ 2 , ] )


/ ( k+1)−sigma∗ ( k∗dt/ ( k +1))ˆ0.5∗z [ 3 , ] )


/ ( k+1)−sigma∗ ( k∗dt/ ( k +1))ˆ0.5∗z [ 4 , ] )


/ ( k+1)−sigma∗ ( k∗dt/ ( k +1))ˆ0.5∗z [ 5 , ] )

s t1<−pmax( s1 . t 1 , s2 . t 1 , s3 . t 1 , s4 . t 1 , s5 . t 1)

s t2<−pmax( s1 . t 1∗ ( s1 . t 1<s t1 ) , s2 . t 1∗ ( s2 . t 1<s t1 ) ,

s3 . t 1∗ ( s3 . t 1<s t1 ) , s4 . t 1∗ ( s4 . t 1<s t1 ) ,

s5 . t 1∗ ( s5 . t 1<s t1 ) )


s3 . t 1∗ ( s3 . t 1<s t2 ) , s4 . t 1∗ ( s4 . t 1<s t2 ) ,

s5 . t 1∗ ( s5 . t 1<s t2 ) )


s3 . t 1∗ ( s3 . t 1<s t3 ) , s4 . t 1∗ ( s4 . t 1<s t3 ) ,

s5 . t 1∗ ( s5 . t 1<s t3 ) )

s t5<−pmin( s1 . t 1 , s2 . t 1 , s3 . t 1 , s4 . t 1 , s5 . t 1)

CE<−pmax( st1 −1 ,0)

idx<−( 1 : ( 2 ∗n ) ) [CE>0]


b a s i s 1<−exp(− s t1 [ idx ] /2)

b a s i s 2<−b a s i s 1∗ s t1 [ idx ]

81

b a s i s 3<−b a s i s 1∗ ( s t1 [ idx ]ˆ2−1)

b a s i s 4<−b a s i s 1∗ ( s t1 [ idx ]ˆ3−3∗ s t1 [ idx ] )

b a s i s 5<−b a s i s 1∗ ( s t1 [ idx ]ˆ4−6∗ ( s t1 [ idx ]ˆ2)+3)

b a s i s 6<−s t2 [ idx ]




ba s i s 10<−b a s i s 6 ˆ2

bas i s 11<−b a s i s 7 ˆ2



bas i s 14<−s t1 [ idx ] ∗b a s i s 6

bas i s 15<−b a s i s 6∗b a s i s 7



bas i s 18<−s t1 [ idx ] ∗b a s i s 6∗b a s i s 7∗b a s i s 8∗b a s i s 9

p<−glm( discountedCC˜b a s i s 1+b a s i s 2+b a s i s 3+b a s i s 4+b a s i s 5+b a s i s 6

+b a s i s 7+b a s i s 8+b a s i s 9+bas i s 10+bas i s 11+bas i s 12

+bas i s 13+bas i s 14+bas i s 15+bas i s 16+bas i s 17

+bas i s 18 )$coef f ic ients

estimatedCC<−p [1 ]+p [ 2 ] ∗b a s i s 1+p [ 3 ] ∗b a s i s 2+p [ 4 ] ∗b a s i s 3+p [ 5 ] ∗b a s i s 4

+p [ 6 ] ∗b a s i s 5+p [ 7 ] ∗b a s i s 6+p [ 8 ] ∗b a s i s 7+p [ 9 ] ∗b a s i s 8

+p [ 1 0 ] ∗b a s i s 9+p [ 1 1 ] ∗bas i s 10+p [ 1 2 ] ∗bas i s 11

+p [ 1 3 ] ∗bas i s 12+p [ 1 4 ] ∗bas i s 13+p [ 1 5 ] ∗bas i s 14

+p [ 1 6 ] ∗bas i s 15+p [ 1 7 ] ∗bas i s 16+p [ 1 8 ] ∗bas i s 17

+p [ 1 9 ] ∗bas i s 18

EF<−rep (0 ,2∗n)

EF[ idx ]=(CE[ idx ]>estimatedCC )


s1 . t<−s1 . t 1 ; s2 . t<−s2 . t 1 ; s3 . t<−s3 . t 1 ; s4 . t<−s4 . t 1 ; s5 . t<−s5 . t 1

}

82

payo f f <− exp(−r∗dt )∗ (CC[ 1 : n]+CC[ ( n+1):(2∗n ) ] ) /2∗K

u s p r i c e<−mean( payo f f )

s . e<−sd ( payo f f )/sqrt (n)

data . frame ( uspr i ce , s . e )

}

A.11. R Codes to Price a European Spread Call

s imul EU spread ca l l<− function (n , s1 , s2 ,K, sigma1 , sigma2 ,

r , dividend ,T, rho ){




z<−L%∗%z

s1<−s1∗exp ( ( r−dividend−1/2∗sigma1 ˆ2)∗T+sigma1∗z [ 1 , ] ∗ (Tˆ 0 . 5 ) )

s2<−s2∗exp ( ( r−dividend−1/2∗sigma2 ˆ2)∗T+sigma2∗z [ 2 , ] ∗ (Tˆ 0 . 5 ) )

CC<−pmax( s1−s2−K, 0 )

payo f f eu<−exp(−r∗T)∗CC


s . e<−sd ( payo f f eu )/sqrt (n)

data . frame ( eupr ice , s . e )

}

83

A.12. R Codes to Price an American Spread Call without Pilot Study for

CV

s imul US spread<−function (n , d , s0 1 , s0 2 ,K, sigma1 , sigma2 ,

dividend1 , dividend2 , r ,T, rho , c ){

dt<−T/d

c o r r e l a t i o n matrix<−matrix (c (1 , rep ( rho , 2 ) , 1 ) ,nrow=2)

L<−t ( chol ( c o r r e l a t i o n matrix ) )


z<−L%∗%z

s1 . t<−s0 1∗exp ( ( r−dividend1−1/2∗sigma1 ˆ2)∗T+sigma1∗ (Tˆ 0 . 5 )∗z [ 1 , ] )


CC<−pmax( s1 . t−s2 . t−K, 0 )


for ( k in (d−1) :1){


z<−L%∗%z

mean<−( log ( s0 1)+k∗log ( s1 . t ) )/ ( k+1)

vo l<−( k∗dt/ ( k +1))ˆ0.5∗z [ 1 , ]

s1 . t 1<−exp(mean+sigma1∗vo l )


vo l<−( k∗dt/ ( k +1))ˆ0.5∗z [ 2 , ]


CE<−pmax( s1 . t 1−s2 . t 1−K, 0 )

idx<−( 1 : n ) [CE>0]


b a s i s 1<−s1 . t 1 [ idx ]

b a s i s 2<−s2 . t 1 [ idx ]


b a s i s 4<−s1 . t 1 [ idx ]ˆ2

b a s i s 5<−s2 . t 1 [ idx ]ˆ2

b a s i s 6<−s1 . t 1 [ idx ]ˆ2∗s2 . t 1 [ idx ]

b a s i s 7<−s1 . t 1 [ idx ] ∗s2 . t 1 [ idx ]ˆ2

84

b a s i s 8<−s1 . t 1 [ idx ]ˆ2∗s2 . t 1 [ idx ]ˆ2


+b a s i s 5+b a s i s 6+b a s i s 7+b a s i s 8 )$coef f ic ients



+p [ 8 ] ∗b a s i s 7+p [ 9 ] ∗b a s i s 8

EF<−rep (0 , n )



s1 . t<−s1 . t 1 ; s2 . t<−s2 . t 1

}

payo f f<−CC∗exp(−r∗dt )

payo f f . c<−payof f−c∗ ( payof feu−k i rk approximation ( s0 1 , s0 2 ,

Q1=1,Q2=1,K,T, b1=r−dividend1 , b2=r−dividend2 ,

r , sigma1 , sigma2 , rho ) )

u s p r i c e . c <− mean( payo f f . c )

se . c<−sd ( payo f f . c )/sqrt (n)

data . frame ( u s p r i c e . c , se . c )

}

A.13. R Codes to Price an American Spread Call with Pilot Study for CV

s imul US spread p i l o t<−function (n , d , s0 1 , s0 2 ,K, sigma1 , sigma2 ,

dividend1 , dividend2 , r ,T, rho ){

n p i l o t<−1000

dt<−T/d



### PILOT STUDY ###


85

z<−L%∗%z



CC<−pmax( s1 . t−s2 . t−K, 0 )


for ( k in (d−1) :1){


z<−L%∗%z


vo l<−( k∗dt/ ( k +1))ˆ0.5∗z [ 1 , ]



vo l<−( k∗dt/ ( k +1))ˆ0.5∗z [ 2 , ]


CE<−pmax( s1 . t 1−s2 . t 1−K, 0 )

idx<−( 1 : n p i l o t ) [CE>0]


b a s i s 1<−s1 . t 1 [ idx ]

b a s i s 2<−s2 . t 1 [ idx ]


b a s i s 4<−s1 . t 1 [ idx ]ˆ2

b a s i s 5<−s2 . t 1 [ idx ]ˆ2




+b a s i s 5+b a s i s 6+b a s i s 7 )$coef f ic ients



EF<−rep (0 , n p i l o t )



s1 . t<−s1 . t 1 ; s2 . t<−s2 . t 1

86

}


c<−cov ( payof feu , payo f f )/var ( payo f f eu )

### PRICING WITH CV ###


z<−L%∗%z



CC<−pmax( s1 . t−s2 . t−K, 0 )


for ( k in (d−1) :1){


z<−L%∗%z


vo l<−( k∗dt/ ( k +1))ˆ0.5∗z [ 1 , ]



vo l<−( k∗dt/ ( k +1))ˆ0.5∗z [ 2 , ]


CE<−pmax( s1 . t 1−s2 . t 1−K, 0 )

idx<−( 1 : n ) [CE>0]


b a s i s 1<−s1 . t 1 [ idx ]

b a s i s 2<−s2 . t 1 [ idx ]


b a s i s 4<−s1 . t 1 [ idx ]ˆ2

b a s i s 5<−s2 . t 1 [ idx ]ˆ2




+b a s i s 5+b a s i s 6+b a s i s 7 )$coef f ic ients

87



EF<−rep (0 , n )



s1 . t<−s1 . t 1 ; s2 . t<−s2 . t 1

}


payo f f . c<−payof f−c∗ ( payof feu−k i rk approximation ( s0 1 , s0 2 ,

Q1=1,Q2=1,K,T, b1=r−dividend1 , b2=r−dividend2 ,

r , sigma1 , sigma2 , rho ) )

u s p r i c e . c <− mean( payo f f . c )

se . c<−sd ( payo f f . c )/sqrt (n)

data . frame ( u s p r i c e . c , se . c )

}

88

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